2009/10/09 by Eugen J. Ionascu, Ionascu, Eugen J., Andrei Markov +1
Mathematics · #11A67 #11D09 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11A67 #msc:11D09
paper · pdf · doi:10.48550/arxiv.0910.1722
Eight pages with seven figures
arxiv created 2009/10/09 · arxiv updated 2009/12/01
Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in \mathbb Z3"), 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 \mathbb Z3. On the other hand, there is a finite (6 or 12) class of regular tetrahedra in \mathbb Z3, associated naturally to each nontrivial solution (a,b,c,d) of the Diophantine equation a2+b2+c2=3d2 and for every nontrivial integer solution (m,n,k) of the equation m2-mn+n2=k2. Every regular tetrahedron in \mathbb Z3 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,\mathbb Q) in terms of the seven integer parameters satisfying the two relations mentioned above.