Folding polyiamonds into octahedra

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2023-01-01 DOI:10.1016/j.comgeo.2022.101917

Eva Stehr, Linda Kleist

引用次数: 0

Abstract

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid – the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exists a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.

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将多胺折叠成八面体

我们研究了折叠成最小但未经研究的柏拉图式固体——八面体的多角体（由三角形网格产生的多边形）。我们展示了许多结果。首先，我们表征了含有正面积孔的可折叠多胺，即除一个外的每个多胺都是可折叠的。其次，我们证明了一个凸的多胺折叠成八面体，当且仅当它包含五个多胺中的一个。第三，我们提出了一个尖锐的尺寸界限：虽然存在尺寸为14的不可折叠的多胺，但每个尺寸至少为15的多胺都折叠成八面体。这清楚地表明，人们可以在多项式时间内测试给定的多胺是否折叠成八面体。最后，我们证明了对于正整数到面的任何赋值，都存在一个折叠成八面体的多多面体，使得覆盖一个面的三角形的数量等于所赋值的数量。

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.