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, this infinite family falls into finitely many similarity classes. While the set of classes is finite, it turns out that a far smaller, stable subset of ``fat'' triangles 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.

翻译：三角形的最长边二分法通过连接其最长边中点与对顶点实现。迭代应用此过程会产生一个无限的三角形族。令人惊讶的是，Stynes（1980）的经典结果表明，对于任意初始三角形，该无限族仅包含有限个相似类。尽管相似类集合有限，但一个更小且稳定的"胖"三角形子集实际上主导了最终网格结构。我们证明了以下渐近面积分布结果：对于每个初始三角形，被终端四元组占据的面积比例趋于一，且收敛速度是指数级的。事实上，我们给出了每一步中三角形的精确分布。我们引入**二分图**并运用谱方法建立了该结果。基于这种主导性，我们完整刻画了具有单个终端四元组的三角形特征，同时构造了具有无界数量终端四元组的三角形序列。此外，我们揭示了终端四元组点的若干基本几何性质，为研究整个轨道的几何分布奠定基础。我们的分析利用了Perdomo与Plaza（2014）的双曲几何框架，并改进了他们的技术方法。