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

TL;DR

This paper establishes a novel connection between primitive two-dimensional words and iterated pedal triangles through symbolic coding, revealing a surprising combinatorial-geometric correspondence.

Contribution

It introduces a finite four-symbol coding of the pedal map and constructs a bijection linking two seemingly unrelated mathematical objects.

Findings

Number of primitive 2D words matches the count of specific pedal triangles.

A finite symbolic coding of the pedal map is constructed.

A bijection between two classes of objects is established.

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.