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.

翻译：三角形的最长边平分（LEB）操作通过连接其最长边的中点与对角顶点实现。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，Adler（1983）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。尽管相似类的集合是有限的，但我们证明了一个更小、稳定的"胖"三角形子集——称为**终端四元组**——实际上主导了最终的网格结构。我们证明了以下渐近面积分布结果：对于每个初始三角形，终端四元组所占面积比例趋于一，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**平分图**并运用谱方法建立了这一结论。基于这种主导性，我们完整刻画了仅具有单个终端四元组的三角形，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了基础。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。