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

Abstract: The Longest Edge Bisection (LEB) 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 Adler (1983) shows that for any initial triangle, this infinite family falls into finitely many similarity classes. While the set of classes is finite, we show that a far smaller, stable subset of ``fat'' triangles, called {\bf terminal quadruples}, effectively dominates the final mesh structure. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by 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 establish 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. Our analysis leverages the hyperbolic geometry framework of Perdomo and Plaza (2014) and refines their techniques.