Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates (“in Z 3 ”), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the same property. To summarize, we show first that there is no regular icosahedron/ dodecahedron in Z 3 . On the other hand, there is a finite (6 or 12) class of regular tetrahedra in Z 3 , associated naturally to each nontrivial solution (a, b, c, d) of the Diophantine equation a 2 + b 2 + c 2 = 3d 2 and for every nontrivial integer solution (m, n, k) of the equation m2 − mn + n 2 = k 2 . Every regular tetrahedron in Z 3 belongs, up to an integer translation and/or rotation, to one of these classes. We then show that each such tetrahedron can be completed to a cube with integer coordinates. The study of regular octahedra is reduced to the cube case via the duality between the two. This work allows one to basically give a description the orthogonal group O(3, Q) in terms of the seven integer parameters satisfying the two relations mentioned above.
Ionascu, Eugen J., "Platonic Solids in Z3" (2011). Faculty Bibliography. 901.