MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock, Josef Tkadlec
{"title":"Deltahedral Domes over Equiangular Polygons","authors":"MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock, Josef Tkadlec","doi":"arxiv-2408.04687","DOIUrl":null,"url":null,"abstract":"A polyiamond is a polygon composed of unit equilateral triangles, and a\ngeneralized deltahedron is a convex polyhedron whose every face is a convex\npolyiamond. We study a variant where one face may be an exception. For a convex\npolygon P, if there is a convex polyhedron that has P as one face and all the\nother faces are convex polyiamonds, then we say that P can be domed. Our main\nresult is a complete characterization of which equiangular n-gons can be domed:\nonly if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the\ninteger edge lengths.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"57 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.04687","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A polyiamond is a polygon composed of unit equilateral triangles, and a
generalized deltahedron is a convex polyhedron whose every face is a convex
polyiamond. We study a variant where one face may be an exception. For a convex
polygon P, if there is a convex polyhedron that has P as one face and all the
other faces are convex polyiamonds, then we say that P can be domed. Our main
result is a complete characterization of which equiangular n-gons can be domed:
only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the
integer edge lengths.
多面体是由单位等边三角形组成的多边形，广义正三角形是每个面都是凸多面体的凸多面体。我们研究的是其中一个面可能是例外的变体。对于凸多边形 P，如果有一个凸多面体以 P 为一个面，而其他所有面都是凸多面体，那么我们就说 P 可以是圆顶的。我们的主要结果是完整地描述了哪些等角 n 边形可以被穹顶化：只有当 n 在 {3, 4, 5, 6, 8, 10, 12} 中，并且只有在边长为整数的某些条件下，这些等角 n 边形才可以被穹顶化。
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