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Ramazan Koc

Publications and source records attributed to Ramazan Koc.

At least 19 recordsLinked to original sources
arXiv2026

Modified Mosseri-Sadoc tiles from $D_6$

A modified set of Mosseri-Sadoc (MS) tiles tessellating 3D Euclidean space with icosahedral symmetry is introduced. The new set of tiles are embedded in dodecahedron with a threefold symmetric order. The modified Mosseri-Sadoc (MMS) tiles can be inflated by a new inflation matrix with positive eigenvalues $\tau^3$ and $\tau$ with the corresponding eigenvectors representing the volumes and the Dehn invariants of the tiles, respectively, where $\tau=\frac{1+\sqrt5}{2}$ is the golden ratio. The MMS tiles are obtained by projection of the 4D and 5D facets of the Delone cells tiling the $D_6$ root lattice in an alternating order. It is also proved that a subset of the lattice $D_6$ projects into the dodecahedron inflated by $\tau^n$ with an arbitrary integer $n$ and tiled by the MMS tiles.

cond-mat.other↗
arXiv2023

Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices $\mathbb{Z}^n$ and Quasicrystallography

Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extensions of the dihedral group $I_2(h)$ of order $2h$, $h=2n$ being the Coxeter number, as a subgroup of affine $B_n$ offers a different perspective to $h$-fold symmetric quasicrystallography. Affine $I_2(h)$ is constructed as the subgroup of the affine $B_n$, the symmetry of the cubic lattice $\mathbb{Z}^n$. The infinite discrete group with local dihedral symmetry of order $2h$ operates on the concentric h-gons obtained by projecting the Voronoi cell of the cubic lattice with $2^n$ vertices onto the Coxeter plane. Voronoi cells tile the space facet to facet, consequently, leading to the tilings of the Coxeter plane with some overlaps of the rhombic tiles. It is noted that the projected Voronoi cell is the overlap of $h$ copy of the $h$-gons tiled with some rhombi and rotated by the angle $2π/h$. After a general discussion on the lattice $\mathbb{Z}^n$ with the affine symmetry $\tilde B_n$ and its affine dihedral subgroup $\tilde I_2(h)$ its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry $\tilde B_n$ $(n=1,2,3,4,5)$ have been presented and shown that the projection of the lattice $B_3$ leads to the hexagonal lattice, the projection of the lattice $B_4$ describes the Ammann-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice $B_5$ describes a quasicrystal structure with local 10-fold symmetry with thick and thin rhombi. It is then straight forward to show that the projections of the cubic lattices with even higher dimensions onto the Coxeter plane may lead to the quasicrystal structures with 12-fold, 18-fold symmetries and so on.

math-ph↗
arXiv2021

From Affine $A_4$ to Affine $H_2$: Group Theoretical Analysis of Five-fold Tilings

The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the $H_2$ symmetry as a subgroup of $A_4$, discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine $H_2$ is obtained as the subgroup of the affine $A_4$. The infinite group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of $A_4$ whose Voronoi cells tessellate the 4D Euclidean space possessing the affine $A_4$ symmetry. It is shown that the projection of the Voronoi cell of the root lattice tiles the Coxeter plane with thick and thin rhombuses with the action of the affine $H_2$ symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths obtained from the projection of the square faces and two types of hexagons obtained from the projection of the hexagonal faces of the Voronoi cell. Structure of the local dihedral symmetry $H_2$ fixing a particular point on the Coxeter plane is determined.

math-ph↗
arXiv2020

Dodecahedral Structures with Mosseri-Sadoc Tiles

3D-facets of the Delone cells representing the deep and shallow holes of the root lattice D6 which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio (tau) with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor tau generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and tau naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths tau to the power n with n equals 3 or greater than 3. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.

math.MG↗
arXiv2020

Icosahedral Tiling with Dodecahedral Structures

Icosahedron and dodecahedron can be dissected into tetrahedral tiles projected from 3D-facets of the Delone polytopes representing the deep and shallow holes of the root lattice D_6. The six fundamental tiles of tetrahedra of edge lengths 1 and τare assembled into four composite tiles whose faces are normal to the 5-fold axes of the icosahedral group. The 3D Euclidean space is tiled face-to-face by the composite tiles with an inflation factor τgenerated by an inflation matrix. The aperiodic tiling is a generalization of the Tubingen triangular tiling in 2-dimensions for the faces of the tiles are made of Robinson triangles. Certain combinations of the tiles constitute dodecahedra with edge lengths of 1 and the golden ratio τ=(1+\sqrt(5))/2.

math.MG↗
arXiv2019

Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly the Delone cells tiling the same root lattice projects onto two isosceles Robinson triangles which also lead to Penrose tilings with kites and darts. We point out that the Coxeter element of order h=n+1 and the dihedral subgroup of order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the Coxeter plane. After setting the general scheme we give examples leading to tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with rhombic and triangular tilings of the plane which are useful in modelling the quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4 fold symmetry.

math.MG↗
arXiv2018

Explicit Construction of the Voronoi and Delaunay Cells of W(An) and W(Dn) Lattices and Their Facets

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(an) and W(dn) are constructed. The face centered cubic (fcc) and body centered cubic (bcc)lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centered at the origin of the root lattice An are the polytopes of the fundamental weights w1, w2, ..., wn and the Delone cells of the root lattice Dn are the polytopes obtained from the weights w1, wn-1, wn. A simple mechanism explains the tessellation of the root lattice by Delone cells. We prove that the (n-1)-facet of the Voronoi cell of the root lattice An is (n-1)-dimensional rhombohedron and similarly the (n-1) -facet of the Voronoi cell of the root lattice Dn is a dipyramid with a base of (n-2)-cube. Volume of the Voronoi cell is calculated via its (n-1) -facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight lattices An* and Dn*.

math.MG↗
arXiv2016

Two groups 2^3.PSL_2(7) and 2^3:PSL_2(7) of order 1344

We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group PSL_2(7).They share the same character table. The group 2^3.PSL_2(7) is a finite subgroup of the Lie Group G_2 preserving the set of octonions \pm e_i , (i=1,2,...,7) representing a 7-dimensional octahedron.Its three maximal subgroups 2^3:7:3, 2^3.S_4 and 4.S_4:2 correspond to the finite subgroups of the Lie groups G_2, SO(4) and SU(3) respectively. The group 2^3:PSL_2(7) representing the split extension possesses five maximal subgroups 2^3:7:3, 2^3:S_4, 4:S_4:2 and two non-conjugate Klein's group PSL_2(7).The character tables of the groups and their maximal subgroups, tensor products and decompositions of the irreducible representations under the relevant maximal subgroups are identified. Possible implications in physics are discussed.

math.GR↗
arXiv2014

12-fold Quasicrystallography from affine F4, B6, and E6

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 technique applicable to any higher dimensional lattice. We point out that the Coxeter number and the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected as well as the dihedral symmetry of the quasicrystal structure. The eigenvectors and eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in nD Euclidean space which lead suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. We give examples for 12-fold symmetric quasicrystal structures obtained by projecting the higher dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6) . These groups share the same Coxeter number h=12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter group represents the rotation of order 12. The canonical projections (strip projections) of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with 4-fold and 6-fold symmetry. We note that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) and are compatible with the experimental results.

math-ph↗
arXiv2014

Group Theoretical Analysis of Quasicrystallography from Projections of Higher Dimensional Lattices Bn

A group theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) has been presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric quasicrystallography. Higher dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank 3 Coxeter subgroups and maximal dihedral subgroups are identified. It has been explicitly shown that when their Voronoi cells are decomposed under the respective rank 3 subgroups W(A3),W(H2) x W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice B4 onto the Coxeter plane represents quasicrystal structures with 8-fold symmetry. The B5 lattice is used to describe the quasicrystals with both 5-fold and 10-fold symmetries. The lattice B6 can describe a 12-fold symmetric quasicrystal as well as a 3D icosahedral quasicrystal depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

math-ph↗
arXiv2013

Affine Wa(A4), Quaternions, and Decagonal Quasicrystals

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 the simple roots of the Coxeter graph I2 (h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group Wa(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

math-ph↗
arXiv2010

Solution of the Bosonic and Algebraic Hamiltonians by using AIM

We apply the notion of asymptotic iteration method (AIM) to determine eigenvalues of the bosonic Hamiltonians that include a wide class of quantum optical models. We consider solutions of the Hamiltonians, which are even polynomials of the fourth order with the respect to Boson operators. We also demonstrate applicability of the method for obtaining eigenvalues of the simple Lie algebraic structures. Eigenvalues of the multi-boson Hamiltonians have been obtained by transforming in the form of the single boson Hamiltonian in the framework of AIM.

math-ph↗
arXiv2010

Remarks on the Solution of the Position Dependent Mass (PDM) Schrödinger Equation

An approximate method is proposed to solve position dependent mass Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.

quant-ph↗
arXiv2010

Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I

In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.

math-ph↗
arXiv2009

Catalan Solids Derived From 3D-Root Systems and Quaternions

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 quaternions. The respective Weyl groups W(A3), W(B3) and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result as the orbits derived from fundamental weights. The Platonic solids are dual to each others however duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), (001) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011) and (111) which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.

math-ph↗
arXiv2008

Solution of the Matrix Hamiltonians via asymptotic iteration method

A method is suggested to obtain solutions of the various quantum optical Hamiltonians in the framework of the asymptotic iteration method. We extend the notion of asymptotic iteration method to solve the 2 \times 2 matrix Hamiltonians. On a particular case, eigenvalues of the Rabi and Rashba Hamiltonians are computed. The method presented here reproduces a number of earlier results in a natural way as well as leads to a novel findings. Possible generalizations of the method are also suggested.

quant-ph↗