### Abstract

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A _{3}, B _{3}, and H _{3} whose simple roots can be represented by quaternions. The respective Weyl groups W(A _{3}), W(B _{3}), and W(H _{3}) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A _{3}), W(B _{3}), and W(H _{3}) by the quaternions simplify the calculations with no reference to the computer calculations.

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

Article number | 051003JMP |

Journal | Journal of Mathematical Physics |

Volume | 51 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2010 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*51*(4), [051003JMP]. https://doi.org/10.1063/1.3356985

**Catalan solids derived from three-dimensional-root systems and quaternions.** / Koca, Mehmet; Ozdes Koca, Nazife; Koç, Ramazan.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 51, no. 4, 051003JMP. https://doi.org/10.1063/1.3356985

}

TY - JOUR

T1 - Catalan solids derived from three-dimensional-root systems and quaternions

AU - Koca, Mehmet

AU - Ozdes Koca, Nazife

AU - Koç, Ramazan

PY - 2010/4

Y1 - 2010/4

N2 - Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.

AB - Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.

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U2 - 10.1063/1.3356985

DO - 10.1063/1.3356985

M3 - Article

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

M1 - 051003JMP

ER -