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- W1658416551 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." @default.
- W1658416551 created "2016-06-24" @default.
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- W1658416551 date "2009-10-09" @default.
- W1658416551 modified "2023-09-27" @default.
- W1658416551 title "Platonic solids in $mathbb Z^3$" @default.
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