A "heron tetrahedron" is one with intger edge lengths, face-areas, and volume. Fred Lunnon and Jan Fricke both conjectured, both based on considerable computer evidence, that every Heron tetrahedron could be rotated & translated so that all its vertices had integer xyz coordinates. Indeed, Lunnon has now found at least one such "integer lattice pose" for every Heron tetrahedron with diameter<100,000. In this post, I will prove their conjecture. Idea behind proof: 3D integer lattice points will be regarded as "lipshitz quaternions" i.e. integer linear combinations of 1,i,j,k, where i*j=-j*i=k, j*k=-k*j=i, k*i=-i*k=j, i^2=j^2=k^2=-1. Fact: Any 3-dimensional (zero real part) Lipshitz quaternion E with integer length |E|, if scaled up by 2 times its length to get 2*|E|*E, becomes a square. Proof: Let R=|E|+E.  Then    R^2 = 2*|E|*E where E can be any pure-imaginary Lipshitz with integer length. QED Conjugation: R~ denotes quaternion conjugate: (A+Bi+Cj+Dk)~=(A-Bi-Cj-Dk). About R as a rotation map    X --> R~ X R: this causes   2*|E|*E = R^2 to map to   R~ R R R = |R|^2 R R = |R|^2 *2*|E|*E which lies on the integer 3D lattice times 2*|E|*|R|^2. Proof of integer pose existence theorem: Initially pose the tetrahedron so 3 of its vertices lie on the 2D integer lattice in the xy plane. (This can be done, as was prove in earlier posts using Gaussian integer GCD as a tool.) The remaining 4th vertex will be called E and is easily seen to have rational xyz coordinates (as a consequence of edge, area, and volume integrality). Let  L=LCM(denominators). Upscale your tetrahedron by L so it is integer posed but nonprimitive. All edge lengths now divisible by L. All vertices but one (namely E) now are on lattice*L and B is on lattice. Compute R=|E|+E. Upscale tet again by 2*|E|. All vertices but one (namely newE=E') now are on lattice*2*|E|*L while E' has been put on lattice*2*|E|. Apply rotation map   X --> R~ X R to all tet vertices X. Now still integer posed, and now E' is on lattice*2*|E|*|R|^2 and all other vertices on lattice*2*|E|*L. Now we can safely de-scale by a factor of 2*|E|*GCD(|R|^2, L) and still everything on integer lattice. But |R|^2=4*|E|^2 is divisible by L since |E|=integer*L. So the de-scaling factor is 2*|B|*L. But that was precisely the net upscaling factor in the first place. So the net effect has been a rotation without any rescaling, and it has made everything integer-lattice-posed. QED. In fact to do this I did not need to have a Heron tetrahedron.  It sufficed if all vertices but one initially Lipshitz, and that one exception initially rational with length=integer. Notice we never even had to use quaternion-GCD anywhere and this proof works in 3D only. Finally, Lunnon & I have reason to suspect that eventually, for very large Heron tetrahedra (even primitive ones) there will often be many integer poses, i.e. arbitrarily high nonuniqueness. This contrasts with the 2D situation for Heron triangles. -- Warren D. Smith http://RangeVoting.org  <-- add your endorsement (by clicking "endorse" as 1st step)