大反棱镜与四元数
Grand Antiprism and Quaternions
摘要：
用单位范数的四元数表示四维半正则多胞形、\textit{grand antiprism}及其400阶对称群的顶点。它遵循\textbf{$E_{8} $}根系的icosian表示，该表示分解为$H_{4} $根系的两个副本。\textit{grand antiprism}的对称性是Coxeter群$W(H_{4})$的极大子群。它是根据Coxeter图$H_{2} \oplus H'_{2}$的20个四元数根构造的群$Aut(H_{2} \oplus H'_{2})$。$H_{4} $的根系由120阶的二元二十面体群\textit{I}表示，构成规则的4D多面体600胞。当从600个单元的顶点中去除对应于图$H_{2} \oplus H'_{2}$的根的20个四元数顶点时，剩余的100个四元数构成\textit{grand antiprism}的顶点。我们给出了一个详细的分析的\textit{grand antiprism}在四元数方面的细胞的建设。构造了\textit{grand antiprism}的对偶多胞形。
英文摘要：
Vertices of the 4-dimensional semi-regular polytope, the \textit{grand antiprism} and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the \textbf{$E_{8} $} root system which decomposes into two copies of the root system of $H_{4} $. The symmetry of the \textit{grand antiprism} is a maximal subgroup of the Coxeter group $W(H_{4})$. It is the group $Aut(H_{2} \oplus H'_{2})$ which is constructed in terms of 20 quaternionic roots of the Coxeter diagram $H_{2} \oplus H'_{2}$. The root system of $H_{4} $ represented by the binary icosahedral group \textit{I}of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram $H_{2} \oplus H'_{2}$ are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the\textit{grand antiprism}. We give a detailed analysis of the construction of the cells of the\textit{grand antiprism} in terms of quaternions. The dual polytope of the \textit{grand antiprism} has been also constructed.
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