Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding

2026-04-29

AI Summary

This work uncovers a profound connection between two-dimensional primitive words and iterated pedal triangles. By introducing a four-symbol finite coding to symbolize the pedal map and employing branched orbit constructions, the authors establish a bijection between $2 \times n$ two-dimensional primitive words over a binary alphabet and the $n$-th order first self-similar pedal triangles. This study constitutes the first precise bridge between combinatorics on words and geometric dynamical systems, rigorously demonstrating an exact correspondence in cardinality between these two classes of objects. In doing so, it unifies symbolic dynamics, combinatorics, and geometric iteration within a coherent framework, offering a novel paradigm for interdisciplinary research at the intersection of discrete mathematics and dynamical geometry.

Abstract

The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes.

Problem

• two-dimensional words
• pedal triangles
• primitive words
• symbolic coding
• combinatorics on words

Innovation

• two-dimensional words
• pedal triangles
• symbolic coding
• bijection
• combinatorics on words