### Abstract

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the four-dimensional semiregular polytope snub 24-cell and its symmetry group (W(D _{4})/C _{2}): S _{3} of order 576 are obtained from the quaternionic representation of the CoxeterWeyl group W(D _{4}). The symmetry group is an extension of the proper subgroup of the CoxeterWeyl group W(D _{4}) by the permutation symmetry of the CoxeterDynkin diagram D _{4}. The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector Λ = (τ, 1, τ, τ) or on the vector Λ = (σ, 1, σ, σ) in the Dynkin basis with τ = 1+√5/2 and σ = 1-√5/2. The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasiregular four-dimensional polytope invariant under the CoxeterWeyl group W(F _{4}). Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24-cell is also constructed. Relevance of these structures to the Coxeter groups W(H _{4}) and W(E _{8}) has been pointed out.

Original language | English |
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Article number | 1250068 |

Journal | International Journal of Geometric Methods in Modern Physics |

Volume | 9 |

Issue number | 8 |

DOIs | |

Publication status | Published - Dec 2012 |

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### Keywords

- quaternions
- Snub 24-cell
- the CoxeterWeyl group W(D )

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*International Journal of Geometric Methods in Modern Physics*,

*9*(8), [1250068]. https://doi.org/10.1142/S0219887812500685