The Takagi Curve and the $β$-Cantor Function from Mechanical Laws

Abstract: This work shows that fractals can be obtained from Mechanical Laws without being forced by any algorithm, closing the gap between the Platonic world of Mathematics and Nature. Fractal tree crown directly emerges when applying elasticity theory to branching stresses in a binary tree. Vertical displacements of nodes are given by the Takagi curve, while the horizontal ones are given by a linear combination of inverses of $\beta$-Cantor functions. In addition, both fractal dimensions are related, which suggests a deeper connection between the Takagi Curve and the $\beta$-Cantor function.