由三维根系和四元数导出的Catalan实体
Catalan Solids Derived From 3D-Root Systems and Quaternions
摘要：
卡塔兰立体是阿基米德立体的顶点，其顶点可以从Coxeter-Dynkin图A3，B3和H3中获得，其简单的根可以用四元数表示。作用在最高权上的各个Weyl群W（A3）、W（B3）和W（H3）产生对应于具有这些对称性的固体的轨道。柏拉图立体和阿基米德立体的顶点是由基本重量导出的轨道。柏拉图立体是彼此对偶的，然而阿基米德立体的第二个是加泰罗尼亚立体，其顶点可以被写为轨道的并集，直到某些比例因子，通过将上述Weyl群应用于每个图的基本最高权（100），（010），（001）获得。这些面由从权重（010）、（110）、（101）、（011）和（111）导出的轨道表示，这些权重对应于阿基米德立体的顶点。用四元数表示Weyl群W（A3），W（B3）和W（H3）简化了计算，而不涉及计算机计算。
英文摘要：
Catalan Solids are the duals of the Archimedean solids, vertices of which can be obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3) and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result as the orbits derived from fundamental weights. The Platonic solids are dual to each others however duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), (001) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011) and (111) which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
