Platonic solids in $\mathbb Z^3$

TL;DR

This paper characterizes all regular polyhedra with vertices in $\\mathbb{Z}^3$, showing the existence of certain tetrahedra related to specific Diophantine equations, and demonstrates how they relate to cubes and the orthogonal group over rationals.

Contribution

It provides a complete classification of regular tetrahedra in $\\mathbb{Z}^3$ and links them to solutions of Diophantine equations, also connecting these to the structure of the orthogonal group.

Findings

No regular icosahedron or dodecahedron in $\\mathbb{Z}^3$.

Finite classes of regular tetrahedra associated with Diophantine solutions.

Every such tetrahedron can be extended to a cube with integer coordinates.

Abstract

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), 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 $\mathbb Z^3$. On the other hand, there is a finite (6 or 12) class of regular tetrahedra in $\mathbb Z^3$, 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 $\mathbb Z^3$ 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 $O(3,\mathbb Q)$ in terms of the seven integer parameters satisfying the two relations mentioned above.