### Abstract

In this series of two papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper, we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of grapheme, where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the regular and quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagram.

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

Pages (from-to) | 41-52 |

Number of pages | 12 |

Journal | African Review of Physics |

Volume | 6 |

Publication status | Published - 2011 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*African Review of Physics*,

*6*, 41-52.

**Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions - I.** / Koca, Mehmet; Koca, Nazife Ozdes; Koc, Ramazan.

Research output: Contribution to journal › Article

*African Review of Physics*, vol. 6, pp. 41-52.

}

TY - JOUR

T1 - Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions - I

AU - Koca, Mehmet

AU - Koca, Nazife Ozdes

AU - Koc, Ramazan

PY - 2011

Y1 - 2011

N2 - In this series of two papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper, we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of grapheme, where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the regular and quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagram.

AB - In this series of two papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper, we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of grapheme, where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the regular and quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagram.

UR - http://www.scopus.com/inward/record.url?scp=84892737462&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892737462&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 41

EP - 52

JO - African Review of Physics

JF - African Review of Physics

SN - 2223-6589

ER -