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

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.