论文标题
新旧几何多面体，几乎没有顶点
Old and new geometric polyhedra with few vertices
论文作者
论文摘要
本文涉及带有顶点标记为通用八面体图的顶点的三角形$ o_4 $，这与完整的四方图$ k_ $ k_ {2,2,2,2,2} $是同构的;众所周知，精确存在十二个这样的三角剖分。我们在超八面体的schlegel图中发现了所有12个三角形，并以3个空间中相同的1个骨骼来几何地实现所有三角形。特别是，我们在3个空间中识别两个几何多面体托里（均无需自我交流），但没有单个共同的脸，或者换句话说，它们的相交（作为点集）只是其共同的1型骨骼。同样，在5-simplex的schlegel图中找到了带有顶点标记的完整图$ k_6 $的2D投影平面的所有十二个三角剖分，并且在4个空间中以相同的1型骨骼在几何上实现；尤其是我们获得了一对Möbius带和一对三角形的投射平面，分别具有相同的1个骨骼（每对内部），分别为3个空间和4个空间，而没有单个公共面。构造的多面体是建模并用地gebra可视化的。
This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such triangulations. We find all the 12 triangulations in a Schlegel diagram of the hyperoctahedron and realize all of them geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (both without self-intersections) with the same 1-skeleton in 3-space, but without a single common face, or in other words their intersection (as point-sets) is only their common 1-skeleton. Similarly, all the twelve triangulations of the 2D projective plane with the vertex labeled complete graph $K_6$ are found in a Schlegel diagram of the 5-simplex and all are realized geometrically with the same 1-skeleton in 4-space; especially we obtain a pair of triangulations of the Möbius band and a pair of triangulated projective planes with the same 1-skeleton (within each pair) in 3-space and 4-space, respectively, without a single common face. The constructed polyhedra are modeled and visualized with GeoGebra.
