Platonic solids in Z3

Text. 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 m(2) - mn + n(2) = k(2). Every regular tetrahedron in Z(3) belongs, up to an integer translation, permutation of the coordinates and symmetries within the planes of coordinates, 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 of the orthogonal group O(3, Q) in terms of the seven integer parameters satisfying the two relations mentioned above. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=DYSpCzrguPM. Published by Elsevier Inc.