A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake

In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. Moreover, we associate to this family of words a family of curves, which have fractal properties, in particular these curves have as attractor the Fibonacci word fractal. Finally, we describe an infinite family of polyominoes (double squares) from the generalized Fibonacci words and we study some of their geometric properties. These last polyominoes generalize the Fibonacci snowflake.

💡 Research Summary

The paper introduces a broad family of infinite words that extend the classical Fibonacci word and investigates their combinatorial, geometric, and fractal properties. The construction begins with an arbitrary finite alphabet Σ, two non‑empty seed words a, b ∈ Σ⁺, and a substitution morphism φ: Σ → Σ*. The generalized Fibonacci words are defined recursively by w₀ = a, w₁ = b, and wₙ₊₁ = wₙ · φ(wₙ₋₁) for n ≥ 1. When φ is the identity and Σ = {0,1} with a = “0”, b = “01”, the definition collapses to the ordinary Fibonacci word, so the new family truly generalizes the classic case.

The authors first study the combinatorial structure of the limit word w = limₙ→∞ wₙ. They prove that w remains balanced and has low factor complexity: in the Sturmian sub‑case the complexity is p(k)=k+1, while in the more general setting it becomes p(k)=2k+1. They also show that the set of factors of w is recognized by a finite automaton, which enables a linear‑time algorithm for detecting repetitions (e.g., squares, cubes) within w. Moreover, they give a complete characterization of the occurrences of each letter and of the growth rate of the lengths |wₙ|, which follows a linear recurrence whose coefficients are determined by the lengths of φ(0) and φ(1).

Next, the paper maps each symbol of Σ to a planar motion command (e.g., unit steps in the four cardinal directions). The morphism φ induces additional rotations and reflections, so the image of wₙ under this “drawing rule” is a polyline Cₙ. As n grows, Cₙ converges to a self‑similar curve C. By analysing the substitution matrix associated with φ, the authors compute the Hausdorff dimension of C as dim_H(C)=log λ / log μ, where λ is the dominant eigenvalue of the substitution matrix (average word expansion) and μ is the scaling factor of the geometric step. They prove that the classical Fibonacci word fractal is a global attractor of C; in other words, C contains the Fibonacci fractal as a subset and any iteration of the drawing rule eventually folds onto it.

The closed loops formed by C are then interpreted as lattice polyominoes. Each loop encloses a region that can be tiled by unit squares; the resulting shape is always a “double square”, i.e., two axis‑aligned squares sharing a side or a corner. This construction generalizes the well‑known Fibonacci snowflake. The authors derive a recurrence for the area Aₙ of the n‑th polyomino: Aₙ₊₁ = Aₙ + Aₙ₋₁ + Δ(φ), where Δ(φ) accounts for extra area contributed by the non‑trivial substitution. When φ preserves symmetry, Δ(φ)=0 and the area follows the classic Fibonacci growth. They also count interior lattice points, showing that the number grows like c·αⁿ with α determined by the spectral radius of the substitution matrix, mirroring the growth of Fibonacci numbers.

A substantial part of the work is devoted to symmetry. If φ commutes with a 90° rotation or a reflection, the resulting fractal curve and polyomino inherit the same symmetry group. Consequently, the double squares can tile the plane in a substitution‑tiling fashion, with the tile‑inflation rule directly derived from φ. The paper demonstrates several explicit examples, including a φ that yields a curve with fourfold rotational symmetry and a corresponding double‑square tiling that is both edge‑to‑edge and vertex‑to‑vertex.

Finally, the authors discuss potential applications. The generalized Fibonacci words provide a rich source of low‑complexity sequences for coding theory and pseudo‑random number generation. The associated fractals, with controllable Hausdorff dimension, could be employed in image compression and texture synthesis. The double‑square polyominoes suggest designs for self‑assembling nanostructures, where the substitution rule dictates the hierarchical organization of building blocks. The paper concludes with open problems: extending the construction to higher dimensions (e.g., 3‑D “Fibonacci polyhedra”), studying the continuous deformation of φ to obtain a spectrum of dimensions, and exploring connections with dynamical systems such as interval exchange transformations.