### Abstract

Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E _{8} root system which decomposes into two copies of the root system of H _{4}. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H _{4}). It is the group Aut(H _{2} ⊕ H′ _{2}) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H _{2} ⊕ H′ _{2}. The root system of H _{4} represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H _{2} ⊕ H′ _{2} are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

Original language | English |
---|---|

Article number | 495201 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 42 |

Issue number | 49 |

DOIs | |

Publication status | Published - 2009 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*42*(49), [495201]. https://doi.org/10.1088/1751-8113/42/49/495201

**Grand antiprism and quaternions.** / Koca, Mehmet; Al Ajmi, Mudhahir; Koca, Nazife Ozdes.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 42, no. 49, 495201. https://doi.org/10.1088/1751-8113/42/49/495201

}

TY - JOUR

T1 - Grand antiprism and quaternions

AU - Koca, Mehmet

AU - Al Ajmi, Mudhahir

AU - Koca, Nazife Ozdes

PY - 2009

Y1 - 2009

N2 - Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E 8 root system which decomposes into two copies of the root system of H 4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H 4). It is the group Aut(H 2 ⊕ H′ 2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H 2 ⊕ H′ 2. The root system of H 4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H 2 ⊕ H′ 2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

AB - Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E 8 root system which decomposes into two copies of the root system of H 4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H 4). It is the group Aut(H 2 ⊕ H′ 2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H 2 ⊕ H′ 2. The root system of H 4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H 2 ⊕ H′ 2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

UR - http://www.scopus.com/inward/record.url?scp=73249131447&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73249131447&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/42/49/495201

DO - 10.1088/1751-8113/42/49/495201

M3 - Article

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 49

M1 - 495201

ER -