2021/12/02 by James East, James E. East, East, James +4
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #math.CO
paper · pdf · doi:10.48550/arxiv.2112.00899
32 pages, 26 figures, 8 tables
arxiv created 2021/12/02 · openalex publication_date 2021/12/02 · arxiv updated 2021/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C≈ 229000.