On the enumeration of integer tetrahedra

TL;DR

This paper investigates the enumeration of integer tetrahedra with fixed perimeter or diameter, highlighting the complexity compared to triangles and proposing classical methods and conjectures about their asymptotic counts.

Contribution

It introduces approaches for enumerating integer tetrahedra, presents computational data, and formulates conjectures on their asymptotic growth patterns.

Findings

Enumeration problem is more complex than for triangles.

Classical orbit enumeration methods are explored.

A conjecture on asymptotic growth of tetrahedra counts is proposed.

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 $n^5/C$ for some constant $C\approx 229000$.