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 - Snub 24-cell
KW - quaternions
KW - the CoxeterWeyl group W(D )
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U2 - 10.1142/S0219887812500685
DO - 10.1142/S0219887812500685
M3 - Article
AN - SCOPUS:84868522083
SN - 0219-8878
VL - 9
JO - International Journal of Geometric Methods in Modern Physics
JF - International Journal of Geometric Methods in Modern Physics
IS - 8
M1 - 1250068
ER -