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

Mehmet Koca, Nazife Ozdes Koca, Muataz Al-Barwani

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article number1250068
JournalInternational Journal of Geometric Methods in Modern Physics
Issue number8
StatePublished - Dec 2012


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

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)


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