On Heron Simplices and Integer Embedding

TL;DR

This paper explores Heron simplices, especially tetrahedra, their relation to integer boxes, and presents a conjecture on their embeddings, with partial proofs and computational verifications in low dimensions.

Contribution

It introduces new results on Heron tetrahedra, links them to integer box problems, and proposes a conjecture on Heron simplex embeddings, supported by computational evidence.

Findings

Heron tetrahedra are connected to integer box existence.

The embedding conjecture is proved in dimension two.

The conjecture is verified computationally in dimension three.

Abstract

In "Unsolved Problems in Number Theory" problem D22 Richard Guy asked for the existence of simplices with integer lengths, areas, volumes... In dimension two this is well known, these triangles are called Heron triangles. Here I will present my results on Heron tetrahedra, their connection to the existence of an integer box (problem D18), the tools for the search for higher dimensional Heron simplices and my nice embedding conjecture about Heron simplices, which I can only proof in dimension two, but I verified it for a large range in dimension three.