$\mathbb Z^3$中的柏拉图立体
Platonic solids in $\mathbb Z^3$
摘要：
扩展以前的结果在空间中的所有等边三角形具有整数坐标的顶点的特征（“在$\mathbb Z^3$“），我们看问题的特征所有正多面体（柏拉图固体）具有相同的属性。总之，我们首先证明了在$\mathbb Z^3$中不存在正二十面体/正十二面体。另一方面，在$\mathbb Z^3$中存在有限（6或12）类正四面体，自然地与丢番图方程$a^2+b^2+c^2=3d^2$的每个非平凡解$(a,b,c,d)$和方程$m^2-mn+n^2=k^2$的每个非平凡整数解$(m,n,k)$相关联。$\mathbb Z^3$中的每一个正四面体都属于这些类中的一个，直到整数平移和/或旋转。然后，我们表明，每个这样的四面体可以完成一个立方体的整数坐标。正八面体的研究通过两者之间的对偶性被简化到立方体的情况。这项工作允许一个基本上给出描述的正交群$O(3,\mathbb Q)$的7个整数参数满足上述两个关系。
英文摘要：
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.
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