Fibonacci Tilings: Substitution & Applications

• Fibonacci tilings are discrete structures using two prototiles with inflation rules from the Fibonacci sequence, producing aperiodic, self-similar patterns.
• They integrate geometric inflation, substitution matrices, and cut-and-project methods to elucidate spectral, dynamical, and combinatorial properties.
• Their applications extend to quasicrystal modeling, photonic materials, and combinatorial enumeration, linking abstract math to physical phenomena.

A Fibonacci tiling is a discrete structure generated by iterative application of substitution (inflation) rules derived from the Fibonacci sequence or its generalizations, resulting in aperiodic order, strong combinatorial identities, and significant ramifications for spectral, dynamical, and physical properties. The canonical construction involves two prototiles (intervals or segments), substitute rules encoding the golden mean, geometric inflation mapping, symbolic combinatorics, cut-and-project lattices, pure-point diffraction, and extensions to higher-dimensional and multivariate systems (Baake et al., 2023).

1. Substitution Rules and Matrix Formalism

The classic one-dimensional Fibonacci tiling uses two prototiles ("short" $S$, "long" $L$; or, algebraically, $b$ and $a$) with the substitution (inflation) rule $\rho$:

$S \mapsto L, \quad L \mapsto LS$

or equivalently $\rho(a)=ab$, $\rho(b)=a$ in the notation of (Baake et al., 2023). This substitution is encoded via the matrix:

$$M = \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}$$

with eigenvalues $\lambda_+ = \phi = \frac{1+\sqrt{5}}{2}$ (the golden ratio) and $\lambda_- = \phi' = \frac{1-\sqrt{5}}{2}$. The left PF eigenvector $(\phi, 1)$ gives natural tile lengths; the right PF eigenvector (proportional to $(\phi^{-1}, \phi^{-2})$) specifies frequencies of tiles. Iteration of $\rho$ yields a bi-infinite, aperiodic tiling sequence.

2. Geometric Realization and Inflation Dynamics

Geometrically, Fibonacci tilings correspond to two intervals $L$ (length $\phi$) and $S$ (length $1$) in $\mathbb{R}$. The inflation map stretches each tile by $\phi$ and subdivides according to the rule: a $L$ tile becomes $L$ + $S$; a $S$ tile becomes a $L$ tile. Formally, denoting prototiles as $T_S$ (length $1$), $T_L$ (length $\phi$):

$\phi\, T_S = T_L, \qquad \phi\, T_L = T_L \cup T_S$

By recursive inflation, any finite patch explodes into a self-similar, aperiodic covering of $\mathbb{R}$ (Baake et al., 2023, Baake et al., 2019).

3. Cut-and-Project Construction and Model Sets

Fibonacci tilings admit a cut-and-project description via the quadratic ring $\mathbb{Z}[\phi]$ and its Minkowski embedding $$L = \{(x, x^*): x \in \mathbb{Z}[\phi]\} \subset \mathbb{R}^2$$, where $x^*$ is algebraic conjugation ($\phi \mapsto 1-\phi$). The tiling is specified by a window $W \subset \mathbb{R}$, often $W = (-1, \phi-1]$, yielding the model set:

$\Lambda = \{x \in \mathbb{Z}[\phi]: x^* \in W\}$

$\Lambda$ precisely indexes the left endpoints of tiles in a Fibonacci tiling (Baake et al., 2023, Baake et al., 2019). By shifting $W$ slightly, one obtains the unique two asymptotic tilings (corresponding to the symbolic two-cycles).

4. Symbolic and Combinatorial Properties

The Fibonacci word is a classic Sturmian sequence of slope $1/\phi^2$, with factor complexity $p(n) = n+1$. The symbolic hull consists of all sequences locally indistinguishable from the fixed point, forming a minimal dynamical system under shift. Notably, factors of fixed length differ in the number of $L$-tiles by at most $1$ (balance property); each finite factor recurs with bounded gaps (repetitivity) (Baake et al., 2023).

Extensions to $k$-Fibonacci recurrences and generalized compositions involve tilings with tiles of length $\leq k$ and satisfy

$F_n^{(k)} = \sum_{i=1}^k F_{n-i}^{(k)}$

with bijections to layered permutations/partitions and $q$-analogues for weighted combinatorial statistics (Goyt et al., 2012, 2609.01838, Hopkins, 1 Sep 2025, Gil et al., 2021).

5. Spectral and Dynamical Properties

A Fibonacci tiling exhibits pure-point diffraction. Its autocorrelation measure is

$$\gamma = \sum_{z \in \Lambda-\Lambda} \nu(z)\, \delta_z$$

with pair-correlation coefficients derived from the covariogram of the cut-and-project window. The diffraction spectrum is

$$\widehat{\gamma} = \sum_{k \in L^\circledast} I(k)\, \delta_k$$

where $$L^\circledast = \frac{1}{\sqrt{5}} \mathbb{Z}[\phi]$$ is the reciprocal module, and Bragg peaks are at wave-numbers $k = m + n\phi$. The hull $Y$ under translation has dynamical spectrum matching the diffraction, with a complete $\mathbb{R}$-indexed set of eigenfunctions (Baake et al., 2023, Baake et al., 2019).

6. Generalizations: Higher Dimensions, Shape Deformations, and Spectra

Fibonacci-type tilings generalize via Pisot substitutions. In $\mathbb{R}^2$, the direct product yields four prototiles. Iterated inflations and rearrangements generate 48 distinct direct product variations (DPVs), all measure-theoretically isomorphic, split into 28 polygonal-window and 20 fractal-window classes. Polygonal DPVs are topologically conjugate by shearing—see classification via MLD classes and cut-and-project sets from lattices in $\mathbb{R}^4$ (Baake et al., 2022, Baake et al., 2019).

Higher-dimensional extensions encompass cubic/quadratic/quasicrystal cases (e.g., Tribonacci, Penrose, Ammann–Beenker), characterized via higher-rank cut-and-project lattices, regular or fractal windows, and pure-point diffraction or mixed spectra. Shape deformations (shear) preserve topological conjugacy; monotile aperiodic cases (Hat, Spectre) arise as non-local shape changes of higher-dimensional model sets (Baake et al., 2023). Cohomological invariants (Anderson–Putnam, pattern-equivariant cohomology) classify deformations and symmetries.

Random substitutions (stochastic letter flips) yield spectra with both pure-point and continuous components. Renormalization relations and window fractality can preclude absolute continuity. The frequency module $\mathbb{Z}[\phi]$ appears systematically in gap-labelling for Schrödinger operators and Meyer set theory (Baake et al., 2023).

7. Applications in Physics, Combinatorics, and Algebra

The Fibonacci tiling is a canonical model for 1D and 2D quasicrystals, photonic/phononic metamaterial spectral theory, ladder graph spanning trees, restricted word and partition enumeration, frieze patterns, and band gap formation in physical systems. The classic tiling encapsulates the minimal complexity, self-similarity, and quasicrystallinity of the golden mean in materials and combinatorics. Quantum transport in separable d-dimensional quasiperiodic tilings exhibits anomalous scaling exponents, RG-invariant coupling transformations, and Fibonacci-related scaling laws (Thiem et al., 2012, Baake et al., 2023).

Generalized Fibonacci tilings define transmission band gaps via Chebyshev-polynomial trace conditions, with rigorous criteria for super band gaps persistent in all periodic approximants of finite quasicrystals (Davies et al., 2023). Hexagonal/trigonal tilings, realized by dual grid or six-dimensional cut-and-project methods, admit combinatorial enumeration of prototile frequencies and substitution rules preserving global 6- or 3-fold symmetry (Coates et al., 2022).

Combinatorial models produce $q$-analogues, colored compositions (given by secondary tilings and ladder graph spanning trees), and Fibonomial coefficients (via F-boxes in $N^\infty$, cobweb posets, and clique problems in graph theory) (M, 2023, Gil et al., 2021, 0802.3473). Higher-order tilings yield identities involving squares/products of Fibonacci numbers, Jacobsthal numbers, Lucas numbers, and binomial-Fibonacci sums (Edwards et al., 2020, Edwards et al., 2019, Silva, 2013).

8. Summary Table: Core Constructs in Fibonacci Tilings

Construct	Mathematical Object	Key Feature
Substitution rule	$S \mapsto L,\; L \mapsto LS$	Encodes inflation/self-similarity
Cut-and-project set	$\{x \in \mathbb{Z}[\phi]: x^* \in W\}$	Generates aperiodic, pure-point structure
Substitution matrix $M$	$\begin{pmatrix}1&1\1&0\end{pmatrix}$	Determines spectral, tile-frequency data
Spectral module	$L^\circledast=\frac{1}{\sqrt{5}}\mathbb{Z}[\phi]$	Diffraction and dynamical eigenfunctions
Generalized recurrence	$F_n^{(k)} = \sum_{i=1}^k F_{n-i}^{(k)}$	$k$-Fibonacci numbers and colored tilings
DPV tilings	4 prototiles, 48 inflation rules	High-dimensional, topological classes
Fibonomial coefficients	$\binom{n}{k}_F = \frac{F_n!}{F_k!F_{n-k}!}$	F-box lattice tilings, cobweb posets

This structure exposes the mathematical richness and connections that underpin Fibonacci tilings, tying symbolic substitutions, geometric inflation rules, cut-and-project formalism, dynamical spectra, combinatorial enumeration, and physical applications in a unified theory (Baake et al., 2023).