  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.
