正多面体的七种不同类型的着色公式

Seven Different Types of Coloring Formulas for Regular Polyhedrons

摘要: 本文我们研究正多面体的顶点、边、面、点边、点面、边面和点边面的着色公式。我们通过考虑正多面体的自同构群(旋转群)在正多面体点、边、面等个体集合上的作用，先求出所有正多面体的旋转群，进而得到这些旋转群在正多面体点、边、面等个体集合上的诱导作用及旋转群中元素的置换类型及个数，然后利用Pólya定理得出正多面体的顶点、边、面、点边、点面、边面和点边面的着色公式。

Abstract: In this paper we obtain the coloring formulas of vertex, edge, surface, point edge, point surface, edge surface, and point edge surface of regular polyhedrons by using Pólya theorem. Our method is to use the knowledge of permutation groups. First we determine the rotation groups of all regular polyhedrons, and then we determine the permutation types and the numbers of elements in the induced permutation groups. Then the formulas are obtained by considering the actions of the groups on the set.