Regular and Irregular chiral polyhedra from Coxeter diagrams via quaternions

Nazife Ozdes Koca, Mehmet Koca

Research output: Contribution to journalArticle

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(A1 ⊕ A1 ⊕ A1), W(A3), W(B3) and W(H3), 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.

Original languageEnglish
Article number148
JournalSymmetry
Volume9
Issue number8
DOIs
Publication statusPublished - Aug 1 2017

Fingerprint

quaternions
Quaternion
polyhedrons
Polyhedron
Irregular
Snub dodecahedron
Diagram
Snub cube
diagrams
Icosahedron
Triangular pyramid
Catalan Solid
tetrahedrons
Dual solid
Archimedean solid
apexes
Coxeter Group
subgroups
Two Parameters
Mirror

Keywords

  • Coxeter diagrams
  • Irregular chiral polyhedra
  • Quaternions
  • Snub cube
  • Snub dodecahedron

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • Mathematics(all)
  • Physics and Astronomy (miscellaneous)

Cite this

Regular and Irregular chiral polyhedra from Coxeter diagrams via quaternions. / Koca, Nazife Ozdes; Koca, Mehmet.

In: Symmetry, Vol. 9, No. 8, 148, 01.08.2017.

Research output: Contribution to journalArticle

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