由Coxeter图和四元数导出的手征多面体
Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions
摘要：
有两种手征阿基米德多面体：短柱立方体和短柱十二面体，以及它们的二分之一：五角二十四面体和五角六十面体。本文系统地构造了手征多面体及其对偶立体。我们利用Coxeter群$W(A_1 \oplus A_1 \oplus A_1)$，$W(A_3)$，$W(B_3)$和$W(H_3)$的转动子群来导出代表感兴趣的固体的轨道。它们分别导致多面体四面体、二十面体、短柱立方体和短柱十二面体。我们证明了四面体和二十面体可以通过适当的旋转八面体群$\frac{W(B_3)}{C_2}$转化为它们的镜像，因此它们不属于手征多面体类。值得注意的是，可以分别通过适当的旋转群$\frac{W(B_3)}{C_2}$和$\frac{W(H_3)}{C_2}$的作用，从作为简单根的非线性组合的向量导出Snub立方体和Snub十二面体。它们的轨道被构造为关注组的三个轨道的并集。我们还构造了多面体，一般准规则，通过结合手征多面体与他们的镜像。作为副积，我们得到了作为Coxeter群$W(H_3)$子群的核多面体群，并讨论了核多面体的构造。我们采用的方法描述的Coxeter群及其轨道的四元数。
英文摘要：
There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their duals the Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups $W(A_1 \oplus A_1 \oplus A_1)$, $W(A_3)$, $W(B_3)$ and $W(H_3)$ to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group $\frac{W(B_3)}{C_2}$ so they are not classified in the class of chiral polyhedra. It is noted that the snub cube and the snub dodecahedron can be derived from the vectors, which are non-linear combinations of the simple roots, by the actions of the proper rotation groups $\frac{W(B_3)}{C_2}$ and $\frac{W(H_3)}{C_2}$ respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by product we obtain the pyritohedral group as the subgroup the Coxeter group $W(H_3)$ and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.
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