A "Heron 4-simplex" is a 4D simplex with all edge lengths, triangle-areas, face-3D-volumes, and 4D-volume all integer. No Heron 4D simplex has ever been found and I suspect none exist. (Can any of you find one? Or prove nonexistence?) But if any do exist, then I now prove that it can be rotated and translated in 4-space so that all its vertices lie on the lattice of "Hurwitz integers" that is so that for each vertex EITHER all 4 coordinates are integers, OR all 4 coordinates are integers+1/2. We shall regard vertices as "Hurwitz quaternions" that is (A,B,C,D) is regarded as A+Bi+Cj+Dk. It is known that Hurwitz quaternion multiplication multiplies lengths, Hurwitzes have only integer lengths (see https://oeis.org/A004011 ) and that there is a GCD algorithm and unique factorization theorem, see book JH Conway & D Smith: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, AK Peters, 2003 Specifically for any factorization F1 of |Q| into ordinary primes in any order, there exists a factorization F2 of the Hurwitz quaternion Q into Hurwitz primes so their lengths are the primes in F1 in the same order. PROOF OF HURWITZ-LATTICE POSABILITY: Use previous 3D posability proof to pose first 4 vertices on 3D integer lattice, then it is easy to see the 5th vertex has rational coordinates in 4D. Let L=LCM(denominators). Upscale your 4simplex by L so it is integer-posed but nonprimitive. All lengths now divisible by L. All vertices but one (namely E) now are on HurwitzLattice*L while E is on HurwitzLattice. E has length=integer hence |E|^2=squared integer, hence due to Hurwitz prime factorization theorem E is factorable within the Hurwitz quaternions E=A*B with |A|^2=|B|^2=|E|=integer. Apply rotation map X --> A~ X B to all 4simplex vertices X. (~ denotes quaternion conjugate, use quaternion multiplication.) Now still integer posed, and now newE = E' = A~ A B B = |A|^2 B^2 is on HurwitzLattice*|A|^2=HurwitzLattice*|E|. Meanwhile all other vertices lie on HurwitzLattice*L. Now we can safely de-scale by a factor of GCD(|E|, L) and still everything on HurwitzLattice. But |E| is divisible by L since |E|=integer*L. So the de-scaling factor is L. But that was precisely the upscaling factor in the first place. So the net effect has been a rotation without any rescaling, and it has made everything HurwitzLattice-posed. QED. NOTES: This theorem only needs to assume at least one edge has integer length and the other vertices (and there need not be 4 others, any number will do) all have integer 4D coordinates, and all the other edges have sqrt(integer) lengths. So it works for more general scenarios than merely Heron 4simplices. (Also my 3D proof worked for the more general scenario where one vertex E has rational xyz coordinates, any number of others have integer xyz coordinates, all edge lengths sqrt(integer) except at least one edge to E has integer length.) Fred Lunnon points out that Jan Fricke gave the example of a regular tetrahedron with edge-length 2 and integer vertices (0, 0, 0, 0), (1, 1, 1, 1), (2, 0, 0, 0), (1, 1, 1, -1). Dividing by 2 gives a set with all distances=1 which is easily seen not be posable on the 4D integer lattice, but it is posed on the HurwitzLattice. This suggests that my use of the HurwitzLattice in my proof (allowing some half-integer coordinates) was essential -- unlike in 3D and 2D where purely integer coordinates suffice for posability of Heron simplices. Finally I want to thank Fred Lunnon whose computer results stimulated all this and helped to restrict our wrong guesses. [It also is possible that a different 3D posability proof exists that is based on Fred Lunnon's computer program for finding poses -- by a semi-empirical process he found a fast pose-finding program that always seems to work, although it is currently not clear why.] -- Warren D. Smith http://RangeVoting.org  <-- add your endorsement (by clicking "endorse" as 1st step)