On the enumeration of integer tetrahedra

IF 0.4 4区 计算机科学 Q4 MATHEMATICS
James East , Michael Hendriksen , Laurence Park
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引用次数: 0

Abstract

We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding problem for triangles, which has long been solved. We expect there are no closed-form solutions to the tetrahedron enumeration problems, but we explore the extent to which they can be approached via classical methods, such as orbit enumeration. We also discuss algorithms for computing the numbers, and present several tables and figures that can be used to visualise the data. Several intriguing patterns seem to emerge, leading to a number of natural conjectures. The central conjecture is that the number of integer tetrahedra of perimeter n, up to congruence, is asymptotic to n5/C for some constant C229000.

关于整数四面体的枚举
我们考虑列举固定周长(边长和)和/或直径(最大边长)的整数四面体,直至相余的问题。正如我们将看到的,这个问题比三角形的相应问题要困难得多,而三角形的相应问题早就解决了。我们期望四面体枚举问题没有封闭形式的解决方案,但我们探索了通过经典方法(如轨道枚举)可以接近的程度。我们还讨论了计算数字的算法,并提供了一些可以用来可视化数据的表格和图表。一些有趣的模式似乎出现了,导致了一些自然的猜想。中心猜想是,对于某常数C≈229000,周长为n的整数四面体的数目渐近于n5/C,直至同余。
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来源期刊
CiteScore
1.60
自引率
16.70%
发文量
43
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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