Lattice Embedding of Heronian Simplices

TL;DR

This paper introduces a novel method using complex and quaternion GCD to embed Heronian simplices into integer lattices, proving uniqueness for triangles and exploring embeddings for tetrahedra with some limitations.

Contribution

It presents a new GCD-based approach for lattice embedding of Heronian simplices, extending known results from triangles to tetrahedra and analyzing uniqueness and limitations.

Findings

Heronian triangles can be embedded uniquely in Z^2

The method extends to tetrahedra using quaternion GCD

Embeddings exist beyond the constructed method, with limitations in higher dimensions

Abstract

A rational triangle has rational edge-lengths and area; a rational tetrahedron has rational faces and volume; either is Heronian when its edge-lengths are integer, and proper when its content is nonzero. A variant proof is given, via complex number GCD, of the previously known result that any Heronian triangle may be embedded in the Cartesian lattice Z^2; it is then shown that, for a proper triangle, such an embedding is unique modulo lattice isometry; finally the method is extended via quaternion GCD to tetrahedra in Z^3, where uniqueness no longer obtains, and embeddings also exist which are unobtainable by this construction. The requisite complex and quaternionic number theoretic background is summarised beforehand. Subsequent sections engage with subsidiary implementation issues: initial rational embedding, canonical reduction, exhaustive search for embeddings additional to those yielded via GCD; and illustrative numerical examples are provided. A counter-example shows that this approach must fail in higher dimensional space. Finally alternative approaches by other authors are summarised.