Unfoldings and nets of regular polytopes

IF 0.4 4区计算机科学 Q4 MATHEMATICS Computational Geometry-Theory and Applications Pub Date : 2023-04-01 DOI:10.1016/j.comgeo.2022.101977

Satyan L. Devadoss , Matthew Harvey

引用次数: 1

Abstract

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for their higher-dimensional analogs, the regular polytopes. Three classes of regular polytopes exist for all dimensions (n-simplex, n-cube, n-orthoplex) and three additional regular polytopes appear only in four-dimensions (24-cell, 120-cell, 600-cell). It was recently proven that all unfoldings of the n-cube yield nets. We extend this to the n-simplex and the 4-orthoplex using the geometry of simplicial chains. Finally, we demonstrate failure of this property for any orthoplex of higher dimension, as well as for the 600-cell, providing counterexamples. We conjecture failure for the two remaining open cases, the 24-cell and the 120-cell.

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正则多面体的展开与网

十多年前，研究表明，柏拉图固体的每一个边缘展开都没有自重叠，从而产生了一个有效的网络。我们考虑它们的高维类似物，规则多面体的这个性质。三类正则多面体存在于所有维度（n-单纯形、n-立方体、n-正交），并且三个额外的正则多面体仅出现在四个维度（24单元、120单元、600单元）。最近证明了n-立方体的所有展开都产生了网络。我们利用单纯形链的几何把它推广到n-单纯形和4-正射算子。最后，我们证明了这种性质在任何高维正射法以及600单元中的失败，并提供了反例。我们推测剩余的两种开放情况，即24细胞和120细胞的失败。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.

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