TY - JOUR
T1 - Quasi regular polyhedra and their duals with coxeter symmetries represented by quaternions -II
AU - Koca, Mehmet
AU - Ajmi, Mudhahir Al
AU - Al-Shidhani, Saleh
PY - 2011
Y1 - 2011
N2 - In this paper, we construct the quasi regular polyhedra and their duals, which are the generalizations of the Archimedean and Catalan solids, respectively. This work is an extension of two previous papers of ours, which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups, W(A3), W(B3) and W (H3). When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups, which necessarily leads to the quaternionic representation of the vertices. We note the fact that the C60 molecule can best be represented by a truncated icosahedron, where the hexagonal faces are not regular. Rather, they are isogonal hexagons with single and double bonds of the carbon atoms represented by the alternating edge lengths of isogonal hexagons.
AB - In this paper, we construct the quasi regular polyhedra and their duals, which are the generalizations of the Archimedean and Catalan solids, respectively. This work is an extension of two previous papers of ours, which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups, W(A3), W(B3) and W (H3). When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups, which necessarily leads to the quaternionic representation of the vertices. We note the fact that the C60 molecule can best be represented by a truncated icosahedron, where the hexagonal faces are not regular. Rather, they are isogonal hexagons with single and double bonds of the carbon atoms represented by the alternating edge lengths of isogonal hexagons.
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M3 - Article
AN - SCOPUS:84860489456
SN - 2223-6589
VL - 6
SP - 53
EP - 67
JO - African Review of Physics
JF - African Review of Physics
ER -