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.

Highlights

  • In fundamental physics, chirality plays a very important role

  • In two earlier publications [8,9], we studied the symmetries of the Platonic–Archimedean solids and their dual solids, the Catalan solids, and constructed their vertices

  • The chiral Archimedean solids, snub cube, snub dodecahedron and their duals have been constructed by employing several other techniques [15,16], but it seems that the method in what follows has not been studied earlier in this context

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Summary

Introduction

Chirality plays a very important role. A Weyl spinor describing a massless. Section we introduce quaternions and construct the Coxeter groups in terms of proper rotational symmetries transforming them to their mirror images. To an icosahedron and again prove that it can be transformed by the group W(B3)+ to its mirror image, In Section 4 we discuss a similar problem for the Coxeter-Dynkin diagram A3 leading to an icosahedron which implies that neither the tetrahedron nor icosahedron are chiral solids. We construct the irregular polyhedra taking the the construction of irregular and regular snub cube and their dual solids from the proper rotational mid-points of edges of the irregular icosahedron as vertices. The chiral irregular and regular snub cube and their dual solids from the proper rotational octahedral symmetry polyhedron taking the mid-points as vertices of the irregular snub cube is discussed.

Quaternionic
The sum of the the face-angles face-angles at at the the vertex vertex Λ
Dual of an Irregular Icosahedron
Regular
An icosidodecahedron icosidodecahedron consists of pentagons of edges
Dual of the Irregular Snub Cube
The proper rotational subgroup of the icosahedral
Factoring by Numbering of the faces has been done according a2
Λirregular
Dual of the Irregular Snub Dodecahedron
Chiral Polyhedra with Vertices at the Edge Mid-Points of the Irregular Snub
Concluding Remarks
Full Text
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