Szajewska Marzena
Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec, Canada.
Acta Crystallogr A Found Adv. 2014 Jul;70(Pt 4):358-63. doi: 10.1107/S205327331400638X. Epub 2014 Jun 11.
This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groups H3, H4. The method consists of recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.
本文研究三维欧几里得空间(R(n))（(3\leq n\lt\infty)）中的柏拉图多面体/多胞形。文中描述了柏拉图多面体/多胞形及其维度为(0\leq d\leq n - 1)的面。同时还平行地考虑了柏拉图多面体的对偶对。其基础的有限考克斯特群是(A(n))型、(B(n))型、(C(n))型、(F_4)型单李代数的考克斯特群，也称为外尔群，或者等价地，晶体学考克斯特群，以及非晶体学考克斯特群(H_3)、(H_4)。该方法包括对适当的考克斯特 - 戴金图进行递归装饰。每个递归步骤都提供了关于特定维度面的基本信息。如果在每个递归步骤中，所有的面都在同一个考克斯特群轨道中，即它们是相同的，那么这个立体就被称为柏拉图式的。本文的主要结果在定理2.1以及命题3.1和3.2中给出。
