# Fibonacci words in hyperbolic Pascal triangles

## TL;DR

This paper explores the relationship between hyperbolic Pascal triangles and Fibonacci words, revealing that certain patterns in these triangles correspond to Fibonacci sequences and generalizing Fibonacci words through geometric structures.

## Contribution

It introduces a novel connection between hyperbolic Pascal triangles and Fibonacci words, providing a geometric interpretation and generalization of Fibonacci sequences.

## Key findings

- Pattern in ${
m HPT}_{4,5}$ matches every second Fibonacci word
- Generalization of Fibonacci words using hyperbolic Pascal triangles
- Graph structure links Fibonacci words with hyperbolic geometry

## Abstract

The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal HPT}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\cal HPT}_{4,q}$ imply a graph structure between the finite Fibonacci words.