Title
Document Type
Article
Publication Date
2011
Publication Title
Elsevier
Abstract
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.
Recommended Citation
Ionascu, Eugen J., "Platonic Solids in Z3" (2011). Faculty Bibliography. 901.
https://csuepress.columbusstate.edu/bibliography_faculty/901
