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Fractal spatio-temporal scale-free messaging: amplitude modulation of self-executable carriers given by the Weierstrass function's components
Published 11 Mar 2024 in cs.IT and math.IT | (2403.06633v5)
Abstract: In many communication contexts, the capabilities of the involved actors cannot be known beforehand, whether it is a cell, a plant, an insect, or even a life form unknown to Earth. Regardless of the recipient, the message space and time scale could be too fast, too slow, too large, or too small and may never be decoded. Therefore, it pays to devise a way to encode messages agnostic of space and time scales. We propose the use of fractal functions as self-executable infinite-frequency carriers for sending messages, given their properties of structural self-similarity and scale invariance. We call it `fractal messaging'. Starting from a spatial embedding, we introduce a framework for a space-time scale-free messaging approach to this challenge. When considering a space and time-agnostic framework for message transmission, it would be interesting to encode a message such that it could be decoded at several spatio-temporal scales. Hence, the core idea of the framework proposed herein is to encode a binary message as waves along infinitely many frequencies (in power-like distributions) and amplitudes, transmit such a message, and then decode and reproduce it. To do so, the components of the Weierstrass function, a known fractal, are used as carriers of the message. Each component will have its amplitude modulated to embed the binary stream, allowing for a space-time-agnostic approach to messaging.
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Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Cocconi, G., Morrison, P.: Searching for interstellar communications. Nature 184(4690), 844–846 (1959) https://doi.org/10.1038/184844a0 Jiang et al. [2022] Jiang, J., Li, H., Chong, M., Jin, Q., Rosen, P., Jiang, X., Fahy, K., Taylor, S., Kong, Z., Hah, J., Zhu, Z.-H.: A beacon in the galaxy: Updated arecibo message for potential fast and seti projects. Galaxies 10(2), 55 (2022) https://doi.org/10.3390/galaxies10020055 H. Zenil [2023] H. Zenil, F.S.A. A. Adams: Agnostic message reconstruction: An optimal and universal decoding method for bio and technosignature detection from first information-theoretic principles arXiv:2303.16045 [cs.IT] (2023) H. Zenil and Rosenblueth [2012] H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Jiang, J., Li, H., Chong, M., Jin, Q., Rosen, P., Jiang, X., Fahy, K., Taylor, S., Kong, Z., Hah, J., Zhu, Z.-H.: A beacon in the galaxy: Updated arecibo message for potential fast and seti projects. Galaxies 10(2), 55 (2022) https://doi.org/10.3390/galaxies10020055 H. Zenil [2023] H. Zenil, F.S.A. A. Adams: Agnostic message reconstruction: An optimal and universal decoding method for bio and technosignature detection from first information-theoretic principles arXiv:2303.16045 [cs.IT] (2023) H. Zenil and Rosenblueth [2012] H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) H. Zenil, F.S.A. A. Adams: Agnostic message reconstruction: An optimal and universal decoding method for bio and technosignature detection from first information-theoretic principles arXiv:2303.16045 [cs.IT] (2023) H. Zenil and Rosenblueth [2012] H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. 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University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. 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Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. 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The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. 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Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. 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Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. 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[2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) H. Zenil, F.S.A. A. Adams: Agnostic message reconstruction: An optimal and universal decoding method for bio and technosignature detection from first information-theoretic principles arXiv:2303.16045 [cs.IT] (2023) H. Zenil and Rosenblueth [2012] H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . 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Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. 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The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. 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Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) H. Zenil, J.A.R.M. C. Gershenson, Rosenblueth, D.: Life as thermodynamic evidence of algorithmic structure in natural environments. Entropy 14(11), 2173–2191 (2012) Seckbach and Gordon [2016] Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Seckbach, J., Gordon, R.: Introduction to biocommunication. In: Gordon, R., Seckbach, J. (eds.) Biocommunication: Sign-mediated Interactions Between Cells And Organisms, p. . World Scientific Publishing, Singapore (2016) Karban [2015] Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Karban, R.: Plant Sensing and Communication. University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. 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Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Seckbach, J., Gordon, R.: Introduction to biocommunication. 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Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. 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University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. 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The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. 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Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. 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RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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University of Chicago Press, Chicago, IL, USA (2015). https://doi.org/10.7208/chicago/9780226264844.001.0001 . https://doi.org/10.7208/chicago/9780226264844.001.0001 Schenk and Seabloom [2010] Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Schenk, H.J., Seabloom, E.W.: In: Baluška, F., Ninkovic, V. (eds.) Evolutionary Ecology of Plant Signals and Toxins: A Conceptual Framework, pp. 1–19. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12162-3_1 Mandelbrot [1982] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco, CA, USA (1982) Falconer [1985] Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. 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Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. 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RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. 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Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. 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The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. 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The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. 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The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. 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Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. 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Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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[2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. 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Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK (1985) Gouyet [1996] Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. 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Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
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Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gouyet, J.F.: Physics and Fractal Structures. Physics and Fractal Structures. Springer, New York, USA (1996) Brooks and Matelski [1981] Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Brooks, R., Matelski, J.P.: In: Kra, I., Maskit, B. (eds.) The Dynamics of 2-Generator Subgroups of PSL(2, C), pp. 65–72. Princeton University Press, Princeton (1981). https://doi.org/10.1515/9781400881550-007 . https://doi.org/10.1515/9781400881550-007 Mandelbrot [1980] Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. Astronomical Society of the Pacific Conference Series, vol. 213, p. 623 (2000) Shen [2017] Shen, W.: Hausdorff dimension of the graphs of the classical weierstrass functions. Mathematische Zeitschrift 289(1–2), 223–266 (2017) https://doi.org/10.1007/s00209-017-1949-1 Lee et al. [2019] Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Mandelbrot, B.B.: Fractal aspects of the iteration of z→λz(1−z)→𝑧𝜆𝑧1𝑧z\to\lambda z(1-z)italic_z → italic_λ italic_z ( 1 - italic_z ) for complex λ𝜆\lambdaitalic_λ and z. Annals of the New York Academy of Sciences 357(1), 249–259 (1980) https://doi.org/10.1111/j.1749-6632.1980.tb29690.x Hardy [1916] Hardy, G.H.: Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society 17(3), 301–325 (1916). Accessed 2024-02-19 Vakoch [2000] Vakoch, D.A.: Three-Dimensional Messages for Interstellar Communication. In: Lemarchand, G., Meech, K. (eds.) Bioastronomy 99. 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The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Lee, D., Watkins, J., Frame, D., Given, G., He, R., Li, N., Lu, B.-N., Sarkar, A.: Time fractals and discrete scale invariance with trapped ions. Physical Review A 100(1) (2019) https://doi.org/10.1103/physreva.100.011403 Rose and Wright [2004] Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Rose, C., Wright, G.: Inscribed matter as an energy-efficient means of communication with an extraterrestrial civilization. Nature 431(7004), 47–49 (2004) https://doi.org/10.1038/nature02884 Gaite [2010] Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Gaite, J.: Fractal analysis of the dark matter and gas distributions in the mare-nostrum universe. Journal of Cosmology and Astroparticle Physics 2010(03), 006–006 (2010) https://doi.org/10.1088/1475-7516/2010/03/006 Falconer [1987] Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Falconer, K.J.: Digital sundials, paradoxical sets, and vitushkin’s conjecture. The Mathematical Intelligencer 9(1), 24–27 (1987) https://doi.org/10.1007/bf03023569 Penrose [1989] Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Inc., USA (1989) Dube [1993] Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Dube, S.: Undecidable problems in fractal geometry. Complex Systems 7(6), 423–444 (1993) Dube [1994] Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994) Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
- Dube, S.: Fractal geometry, turing machines and divide-and-conquer recurrences. RAIRO - Theoretical Informatics and Applications - Informatique Theorique et Applications 28(3-4), 405–423 (1994)
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