发表机构
Bolyai Institute, University of Szeged（塞格德大学博莱伊研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文报告了第二个嵌入三维空间中的八面体多面体，其每两个面共享至少一条边，具有八面九边形、24顶点、36边，与已知例子组合不等价，且具有D2对称性。
AI 中文摘要
我们报告了一个嵌入在 $\mathbb{R}^3$ 中的亏格为 3 的新型多面体曲面，它有 8 个平面、简单、非凸的九边形面，24 个顶点和 36 条边，其中每两个面至少共享一条边：20 对面共享一条边，8 对面共享两条共线边。其面所在的平面为 $3x-4y-2z=5$、$-2x+5y-5z=3$ 以及它们关于三个坐标轴半周转动的像，所有顶点都是有理点。该多面体与 Mizhaev 描述的多面体具有相同的面向量、面大小和边重数，但与之并非组合等价。我们的实现具有阶为 4 的对称群 $D_2$，而 Mizhaev 的多面体具有旋转反射对称性。该例子是通过计算几何搜索发现的，其所有性质均在精确有理算术中得到了验证。
英文摘要
We report a novel polyhedral surface of genus~3 embedded in $\mathbb{R}^3$ with eight planar, simple, non-convex nonagonal faces, 24 vertices and 36 edges, in which every two faces share at least one edge: 20 pairs of faces share one edge and 8 pairs share two collinear edges. Its face planes are $3x-4y-2z=5$, $-2x+5y-5z=3$ and their images under the half-turns about the three coordinate axes, and all vertices are rational. The polyhedron has the same face vector, face sizes and number of edge multiplicities as the polyhedron described by Mizhaev, but it is not combinatorially equivalent to it. Our realisation has the symmetry group $D_2$ of order~4, whereas Mizhaev's polyhedron has a rotoreflection symmetry. The example was found by a computational geometric search, and all its properties were verified in exact rational arithmetic.
