A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake

TL;DR

This paper introduces a generalized family of infinite words extending Fibonacci words, studies their fractal curves and geometric properties, and generalizes the Fibonacci snowflake through polyominoes.

Contribution

It presents a new family of infinite words, explores their fractal curves, and generalizes the Fibonacci snowflake with associated polyominoes.

Findings

Fractal curves with Fibonacci attractors

Generalized Fibonacci snowflake polyominoes

New combinatorial properties of the words

Abstract

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.