Enumeration of integral tetrahedra

TL;DR

This paper presents a comprehensive enumeration of integral tetrahedra up to diameter 1000, providing algorithms for enumeration and canonicity checking, and deriving an exact formula for related matrix classes.

Contribution

It introduces algorithms for enumerating integral tetrahedra and checking their canonicity, along with an exact formula for certain matrix classes, advancing computational geometry methods.

Findings

Enumerated integral tetrahedra up to diameter 1000.

Developed an $O(d^5)$ enumeration algorithm.

Derived an exact formula for specific matrix classes.

Abstract

We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula.