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.

翻译：三角形的最长边二分法通过连接最长边中点与对应顶点实现。迭代应用此过程将生成无穷三角形族。令人惊讶的是，Stynes（1980）的经典结果表明，对任意初始三角形，该无穷族中的元素仅属于有限个相似类。尽管相似类集合有限，但更小规模的周期性"胖"三角形子集实际上主导了最终网格结构。该子集由长度为四的周期轨道构成，我们称之为**终端四元组**。我们证明如下渐近面积分布结果：对每个初始三角形，这些终端四元组所占面积比例趋近于1，且收敛呈指数速率。实际上，我们给出了每个步骤中三角形的精确分布。引入**二分图**并运用谱方法证明该结论。鉴于这种主导性，我们完整刻画了仅含单个终端四元组的三角形特征，并反向构建了具有无界多个终端四元组的三角形序列。此外，我们揭示了终端四元组点集若干基本几何性质，为研究整个轨道的几何分布奠定基础。