Next Article in Journal
Gate Antiphase of Potassium Channel
Previous Article in Journal
Research on Lower Limb Motion Recognition Based on Fusion of sEMG and Accelerometer Signals
Previous Article in Special Issue
The Roundest Polyhedra with Symmetry Constraints
Article Menu

Export Article

Open AccessArticle
Symmetry 2017, 9(8), 148; doi:10.3390/sym9080148

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: mehmetkocaphysics@gmail.com
*
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)

Abstract

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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

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

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top