Polyhedra obtained from Coxeter groups and quaternions

Koca, Mehmet; Ajmi, Mudhahir Al; Koc, Ramazan

doi:10.1063/1.2809467

Cited by 25 publications

(40 citation statements)

References 18 publications

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“…The orbit describing the vertices of the icosidodecahedron is given by [11] 144 .The Catalan solid, rhombic triacontahedron is not only face transitive but also edge transitive. The centers of the edges of the rhombus ABCD are given by the set of quaternions…”

Section: Rhombic Triacontahedron ( Dual Of the Icosidodecahedron)mentioning

confidence: 99%

“…We have discussed the Archimedean solids in reference [11] possessing the icosahedral symmetry . They describe respectively the Archimedean solids, icosidodecahedron, truncated dodecahedron, truncated icosahedron, small rhombicosidodecahedron, and great rhombicosidodecahedron.…”

Section: () Whmentioning

confidence: 99%

“…Consequently, the Coxeter-Weyl groups were expressed using the binary tetrahedral, binary octahedral and the binary icosahedral subgroups of quaternions. In this paper we construct the vertices of the Catalan solids, duals of the Archimedean solids, using the same technique described in the paper [11]. In section 2 we set up the general frame by summarizing the technique of ref.…”

Section: Introductionmentioning

confidence: 99%

“…In ref. [11] we have studied, in detail, the Platonic and the Archimedean solids employing the above technique where the simple roots are described by quaternions. Consequently, the Coxeter-Weyl groups were expressed using the binary tetrahedral, binary octahedral and the binary icosahedral subgroups of quaternions.…”

Section: Introductionmentioning

confidence: 99%

“…In 4D, in addition to the Platonic Polytopes described by the symmetries of . The 24-cell is said to be selfdual because the 4 F diagram is invariant under the Dynkin-diagram symmetry [9,10].…”

Section: Introductionmentioning

confidence: 99%

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

Koca

Koc

2010

Journal of Mathematical Physics

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Catalan Solids are the duals of the Archimedean solids, vertices of which can be obtained from the Coxeter-Dynkin diagrams 3 () WH acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result as the orbits derived from fundamental weights. The Platonic solids are dual to each others however 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), (001) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011) and (111)

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Section: Rhombic Triacontahedron ( Dual Of the Icosidodecahedron)mentioning

confidence: 99%

Section: () Whmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Koca

Koc

2010

Journal of Mathematical Physics

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Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

Dechant

2012

Adv. Appl. Clifford Algebras

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Abstract. Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E 8 , have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A 1 × A 1 × A 1 , A 3 , B 3 and H 3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A 1 × A 1 × A 1 , A 3 , B 3 and H 3 yield a new simple construction of binary polyhedral groups. These in turn generate the groupsand H 4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D 4 , F 4 and H 4 are exceptional structures, which our study sheds new light on.IPPP/12/26, DCPT/12/52Mathematics Subject Classification (2010). Primary 51F15, 20F55; Secondary 15A66, 52B15.

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A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Dechant

2013

Adv. Appl. Clifford Algebras

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Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D 4 , F 4 and H 4 . A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I 2 (n) and the three-dimensional groups A 3 , B 3 , as well as the icosahedral group H 3 .

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Polyhedra obtained from Coxeter groups and quaternions

Cited by 25 publications

References 18 publications

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

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

Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

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