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Affine subgroups of the affine Coxeter group with the same Coxeter number
Nazife Ozdes Koca, Mehmet Koca
Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn)…
Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices and Quasicrystallography
Nazife Ozdes Koca, Mehmet Koca, Ramazan Koc +1
Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extensions of the dihedral group of order , being the…
4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices
Mehmet Koca, Nazife Ozdes Koca, Amal Juma Hamood Al-Qanobi
We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgro…
Quaternionic Representations of the Pyritohedral Group, Related Polyhedra and Lattices
Nazife Ozdes Koca, Mehmet Koca, Aida Al-Mukhaini +1
We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine Coxeter groups W(D3) and W(B3…
12-fold Quasicrystallography from affine F4, B6, and E6
Nazife O. Koca, Mehmet Koca, Ramazan Koc
One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general t…
Affine Wa(A4), Quaternions, and Decagonal Quasicrystals
Mehmet Koca, Nazife O. Koca, Ramazan Koc
We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by t…
Radii of the E8 Gosset Circles as the Mass Excitations in the Ising Model
Mehmet Koca, Nazife Ozdes Koca
The Zamolodchikov's conjecture implying the exceptional Lie group E8 seems to be validated by an experiment on the quantum phase transitions of the 1D Ising model carried out by th…
Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)
Mehmet Koca, Mudhahir Al-Ajmi, Nazife Koca
4-dimensional polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group where the group elements and the vertices of the…
Branching of the W(H4) Polytopes and Their Dual Polytopes under the Coxeter Groups W(A4) and W(H3) Represented by Quaternions
Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi
4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes a…
Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)
Mehmet Koca, Nazife Ozdes Koca, Muataz Al-Barwani
Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the 4-dimensional semi-regular polytop…
4D-Polytopes and Their Dual Polytopes of the Coxeter Group Represented by Quaternions
Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi
4-dimensional polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group where the group elements and the vertices of the p…
Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions
Mehmet Koca, Nazife Ozdes Koca, Muna Al-Shu'eili
There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their duals the Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahe…
Catalan Solids Derived From 3D-Root Systems and Quaternions
Mehmet Koca, Nazife Ozdes Koca, Ramazan Koc
Catalan Solids are the duals of the Archimedean solids, vertices of which can be obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots can be represented by qu…
Grand Antiprism and Quaternions
Mehmet Koca, Mudhahir Al-Ajmi, Nazife Ozdes Koca
Vertices of the 4-dimensional semi-regular polytope, the \textit{grand antiprism} and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It fol…
Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E_8 Root System
Mehmet Koca, Mudhahir Al-Ajmi, Nazife Ozdes Koca
Vertices of the 4-dimensional semi-regular polytope, \textit{snub 24-cell} and its symmetry group of order 576 are represented in terms of quaternions with unit n…