四元数下的4维五角十二面体对称性、相关多胞体与格
4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices
AI总结：
本文用四元数将五角十二面体对称性推广到4维，证明其为W(F4)和W(H4)的极大子群，并构造了相关的伪扭棱24胞体及其对偶，其极限对应黄金比例，相关格为W(D4)根格。
摘要：
我们描述了五角十二面体对称性向4维欧几里得空间的推广，并以四元数形式给出了群元素。结果表明，该群是秩为4的Coxeter群W(F4)和W(H4)两者的极大子群，这意味着该群与4维晶体学以及准晶体结构都相关。首先，我们回顾了3维欧几里得空间中的五角十二面体对称性，它是Coxeter-Weyl群W(B3)=Aut(D3)和W(H3)两者的极大子群。在3维空间中，相关的多面体是两个对偶多面体：伪二十面体-五角十二面体和伪二十-十二面体。在四元数表示下，该对称性自然地推广到4维。相关的多胞体被证明是伪扭棱24胞体及其对偶多胞体，它们用参数x表示；当参数x取黄金比例时，在极限情况下得到扭棱24胞体及其对偶。结果表明，相关的格是W(D4)的根格。
英文摘要：
We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4) implying that it is a group relevant to the crystallography as well as quasicrystallographic structures in 4-dimensions. First we review the pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron- pyritohedron and the pseudo icosidodecahedron. In quaternionic representations it finds a natural extension to the 4-dimensions.The related polytopes turn out to be the pseudo snub 24-cell and its dual polytope expressed in terms of a parameter x leading to snub 24-cell and its dual in the limit where the parameter x takes the golden ratio. It turns out that the relevant lattice is the root lattice of W(D4).
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