新柏拉图立体
Neoplatonic solids
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中文总结 AI 辅助
该研究定义了新柏拉图立体，通过计算机辅助证明等方法，明确了素6-网的理想与近似欧氏新柏拉图体实现，还扩展了欧氏新柏拉图体存在性的适用范围。
中文摘要 AI 辅助
6-网是指最大度不超过6的2-球面单纯三角剖分，实验表明每个6-网都存在唯一的两种实现：一种是由单位等边三角形构成的无凹陷欧氏多面体，另一种是理想等边双曲多面体，我们将这两类分别称为新柏拉图立体和理想新柏拉图体。若一个网的每个3-环都界定一个面，则称其为素网。一项计算机辅助证明显示，每个顶点数v≤50的素6-网都有唯一的凸理想新柏拉图体实现；通过该实现的数值同伦可得到近似欧氏新柏拉图体，另一项计算机辅助证明表明存在真实的欧氏新柏拉图体位于其附近，但未证明唯一性。利用分离三角分解，我们将欧氏存在性扩展到所有顶点数v≤50、按组合同构计数的10412340个6-网。
英文摘要
A \emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \emph{neoplatonic solids} and \emph{ideal neoplatonics}. A net is \emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realization yields an approximate Euclidean neoplatonic, and a computer-assisted proof shows that a true Euclidean neoplatonic lies nearby, though we do not prove uniqueness. Using the separating-triangle decomposition, we extend Euclidean existence to all $10{,}412{,}340$ $6$-nets with $v\leq50$, counted up to combinatorial isomorphism.
