On the stability, complexity, and distribution of similarity classes of the longest edge bisection process for triangles

TL;DR

This paper analyzes the long-term behavior of the longest edge bisection process for triangles, revealing that a small set of 'fat' triangles dominates the final mesh structure with exponential convergence.

Contribution

It proves the dominance of periodic 'fat' triangles in the bisection process and characterizes triangles with a single terminal quadruple, advancing understanding of mesh generation.

Findings

Terminal quadruples occupy almost all the area as the process progresses.

The distribution of triangles in the process converges exponentially.

Complete characterization of triangles with a single terminal quadruple.

Abstract

The Longest Edge Bisection of a triangle is performed by joining the midpoint of its longest edge to the opposite vertex. Applying this procedure iteratively produces an infinite family of triangles. Surprisingly, a classical result of Stynes (1980) shows that for any initial triangle, the elements of this infinite family fall into finitely many similarity classes. While the set of classes is finite, it turns out that a far smaller, periodic subset of ``fat'' triangles effectively dominates the final mesh structure. This subset is comprised of periodic orbits of length four, which we refer to as {\bf terminal quadruples}. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by these terminal quadruples tends to one, with the convergence occurring at an exponential rate. In fact, we provide the precise distribution of triangles in every step. We introduce the {\bf bisection graph} and use spectral methods to prove this result. Given this dominance, we provide a complete characterization of triangles possessing a single terminal quadruple, while conversely exhibiting a sequence of triangles with an unbounded number of terminal quadruples. Furthermore, we reveal several fundamental geometric properties of the points of a terminal quadruple, laying the groundwork for studying the geometric distribution of the entire orbit.