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Symmetry 2017, 9(8), 148;

Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions

Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Oman
Retired professor:
Author to whom correspondence should be addressed.
Received: 6 July 2017 / Revised: 1 August 2017 / Accepted: 2 August 2017 / Published: 7 August 2017
(This article belongs to the Special Issue Polyhedral Structures)


Vertices and symmetries of regular and irregular chiral polyhedra are represented by quaternions with the use of Coxeter graphs. A new technique is introduced to construct the chiral Archimedean solids, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexecontahedron. Starting with the proper subgroups of the Coxeter groups W ( A 1 A 1 A 1 ) , W ( A 3 ) , W ( B 3 ) and W ( H 3 ) , we derive the orbits representing the respective solids, the regular and irregular forms of a tetrahedron, icosahedron, snub cube, and snub dodecahedron. Since the families of tetrahedra, icosahedra and their dual solids can be transformed to their mirror images by the proper rotational octahedral group, they are not considered as chiral solids. Regular structures are obtained from irregular solids depending on the choice of two parameters. We point out that the regular and irregular solids whose vertices are at the edge mid-points of the irregular icosahedron, irregular snub cube and irregular snub dodecahedron can be constructed. View Full-Text
Keywords: Coxeter diagrams; irregular chiral polyhedra; quaternions; snub cube; snub dodecahedron Coxeter diagrams; irregular chiral polyhedra; quaternions; snub cube; snub dodecahedron

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Koca, N.O.; Koca, M. Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions. Symmetry 2017, 9, 148.

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