Group theoretical analysis of 600-cell and 120-cell 4D polytopes with quaternions

Abstract

cell {3, 3, 5} and 120-cell {5, 3, 3} four-dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the Coxeter group W( H4). The maximal subgroups W( SU( 5)) : Z2 and W( H3) × Z2 of W( H4) play important roles in the analysis of cell structures of the dual polytopes. In particular, the Weyl- Coxeter group W( SU( 4)) is used to determine the tetrahedral cells of the polytope {3, 3, 5}, and the Coxeter group W( H3) is used for the dodecahedral cells of {5, 3, 3}. Using the Lie algebraic techniques in terms of quaternions, we explicitly construct cell structures forming the vertices of the 4D polytopes.