### 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 |
---|---|

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

**Snub 24-cell derived from the Coxeter-Weyl group W(D 4).** / Koca, Mehmet; Koca, Nazife Ozdes; Al-Barwani, Muataz.

Research output: Contribution to journal › Article

*International Journal of Geometric Methods in Modern Physics*, vol. 9, no. 8, 1250068. https://doi.org/10.1142/S0219887812500685

}

TY - JOUR

T1 - Snub 24-cell derived from the Coxeter-Weyl group W(D 4)

AU - Koca, Mehmet

AU - Koca, Nazife Ozdes

AU - Al-Barwani, Muataz

PY - 2012/12

Y1 - 2012/12

N2 - 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.

AB - 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.

KW - quaternions

KW - Snub 24-cell

KW - the CoxeterWeyl group W(D )

UR - http://www.scopus.com/inward/record.url?scp=84868522083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868522083&partnerID=8YFLogxK

U2 - 10.1142/S0219887812500685

DO - 10.1142/S0219887812500685

M3 - Article

AN - SCOPUS:84868522083

VL - 9

JO - International Journal of Geometric Methods in Modern Physics

JF - International Journal of Geometric Methods in Modern Physics

SN - 0219-8878

IS - 8

M1 - 1250068

ER -