Nazife Koca | Sultan Qaboos University (original) (raw)

Papers by Nazife Koca

International Journal of Geometric Methods in Modern Physics, Apr 1, 2014

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional ... more 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 2 () Ih where h is the Coxeter number of the Coxeter group () WGwhich embeds the dihedral group h D of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A 4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine 4 () a WAcan be decomposed into two orthogonal spaces whose point groups is the dihedral group 5 D 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

arXiv (Cornell University), Feb 17, 2023

Quasicrystals described as the projections of higher dimensional cubic lattices, and the particul... more Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extension (2 (ℎ)) of the dihedral group (2 (ℎ)) of order 2h, ℎ = 2 being the Coxeter number, as a subgroup of affine group () offers a different perspective to h-fold symmetric quasicrystallography. The affine group (2 (ℎ)) is constructed as the subgroup of the affine group (), the symmetry of the cubic lattice ℤ. 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 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 copies of the h-gons tiled with some rhombi and rotated by the angle 2 ℎ. After a general discussion on the lattice ℤ with its affine group () embedding the affine dihedral group (2 (ℎ)) as a subgroup, its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry (), (= 1,2,3,4,5) have been presented and shown that the projection of the lattice ℤ 3 leads to the hexagonal lattice, the projection of the lattice ℤ 4 describes the Amman-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice ℤ 5 describes a quasicrystal structure with local 10fold 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.

arXiv (Cornell University), Nov 1, 2021

The projections of the lattices, may be used as models of quasicrystals, and the particular affin... more The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the 2 symmetry as a subgroup of 4 , discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine 2 is obtained as the subgroup of the affine 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell (0) of 4 project into the decagonal orbit of 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of 2. It is shown that the thick and thin rhombuses constitute the finite-fragments of the tiles of the Coxeter plane with the action of the affine 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 2 fixing a particular point on the Coxeter plane is determined

Acta Crystallographica, Apr 25, 2022

The projections of the lattices, may be used as models of quasicrystals, and the particular affin... more The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the 2 symmetry as a subgroup of 4 , discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine 2 is obtained as the subgroup of the affine 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell (0) of 4 project into the decagonal orbit of 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of 2. It is shown that the thick and thin rhombuses constitute the finite-fragments of the tiles of the Coxeter plane with the action of the affine 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 2 fixing a particular point on the Coxeter plane is determined

arXiv (Cornell University), Jun 8, 2016

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present t... more 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 subgroup of both the rank-4 Coxeter groups 44 4

Journal of physics, Nov 26, 2014

We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter... more We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter-Weyl group () an WB. An h-fold symmetric quasicrystal structure follows from the hyperqubic lattice whose point group is described by the Coxeter-Weyl group () n WB with the Coxeter number h=2n. Higher dimensional cubic lattices are explicitly constructed for 4,5,6 n  by identifying their rank-3 Coxeter subgroups and maximal dihedral subgroups. Decomposition of their Voronoi cells under the respective rank-3 subgroups 3 2 1 (), () () W A W H W A  and 3 () WH lead to the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice 4 B describes a quasicrystal structure with 8-fold symmetry. The 5 B lattice leads to quasicrystals with both 5fold and 10 fold symmetries. The lattice 6 B projects on a 12-fold symmetric quasicrystal as well as a 3D icosahedral quasicrystal depending on the choice of subspace of projections. The projected sets of lattice points are compatible with the available experimental data.

Journal of physics, Mar 1, 2011

4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl g... more 4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl groups. Their symmetries follow from the quaternionic descriptions of the rank-4 Coxeter-Dynkin diagrams. There exists a one to one correspondence between the finite subgroups of quaternions and the rank-4 Coxeter-Weyl groups. 1.

Journal of Physics: Conference Series

The present work offers a different perspective for the 5-fold symmetric quasicrystallography by ... more The present work offers a different perspective for the 5-fold symmetric quasicrystallography by employing affine H 2 as a subgroup of affine A 4. It is shown that the projection of the Voronoi cell of the root lattice A 4 can be dissociated as identical five decagons up to a rotation tiled by thick and thin rhombuses. 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 and two types of hexagons. Structure of the local dihedral symmetry H 2 fixing a particular point on the Coxeter plane is determined.

Acta Crystallographica Section A Foundations and Advances

The 3D facets of the Delone cells of the root latticeD6which tile the 6D Euclidean space in an al... more The 3D facets of the Delone cells of the root latticeD6which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 51/2)/2 with faces normal to the fivefold and threefold 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 from sixfundamentaltiles, the faces of which are normal to the fivefold 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 τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries...

Journal of Physics: Conference Series, 2014

One can obtain the quasicrystallographic structures from the projection of the higher dimensional... more One can obtain the quasicrystallographic structures from the projection of the higher dimensional lattices into 2D or 3D subspaces. Here we introduce a general technique applicable to any higher dimensional lattice described by the affine Coxeter groups. It is pointed out that the Coxeter number h and the Coxeter exponents play an important role in determining the principal planes onto which the lattice to be projected. The quasicrystal structures obtained by projection display the dihedral symmetry of order 2h . The projection subspaces are determined by using the eigenvectors and the corresponding eigenvalues of the Cartan matrix. Examples are given for 12-fold symmetric quasicrystal structures obtained by projections of the lattices determined by the affine Coxeter-Weyl groups Wa (F4), Wa (B6), and Wa (E6). The reflection generators R1 and R2 of the dihedral group D12 can be obtained as the products of the generators of the Coxeter-Weyl groups. It is noted that the quasicrystal structures obtained from the lattices Wa (F4) and Wa (B6) are compatible with the experimental data.

arXiv (Cornell University), Jun 8, 2016

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present t... more 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 subgroup of both the rank-4 Coxeter groups 44 4

Acta Crystallographica Section A Foundations and Advances, 2021

3D-facets of the Delone cells of the root lattice which tile the six-dimensional Euclidean space ... more 3D-facets of the Delone cells of the root lattice 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 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 generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and 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 with n th power of the golden ratio. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.

arXiv: Mathematical Physics, 2015

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) latti... more 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)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices have been constructed in terms of quaternions. Reflection planes of the Coxeter-Dynkin diagrams are identified with certain planes of the unit cube. It turns out that the pyritohedral symmetry takes a simpler form in terms of quaternionic representation. The D3 diagram is used to construct the vertices of polyhedra relevant to the cubic lattices and, in particular, constructions of the pseudoicosahedron and its dual pyritohedron are explicitly worked out.

SSRN Electronic Journal, 2019

Main motivation for this work is to evaluate students’ understanding of force and motion concepts... more Main motivation for this work is to evaluate students’ understanding of force and motion concepts in two calculus based introductory physics courses at Sultan Qaboos University, Muscat, Oman by using Force Concept inventory (FCI).

International Journal of Geometric Methods in Modern Physics, 2018

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see... more We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra projec...

Sultan Qaboos University Journal for Science [SQUJS], 2019

We analyze the group structures of two groups of order 1344 which are respectively non-split and ... more 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 2 (7). Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal. The group 2 3. 2 (7) is a finite subgroup of the Lie Group 2 preserving the set of octonions ± , (= 1,2, … ,7) representing a 7dimensional octahedron. Its three maximal subgroups 2 3 : 7: 3, 2 3. 4 and 4. 4 : 2 correspond to the finite subgroups of the Lie groups 2 , (4) and (3) respectively. The group 2 3 : 2 (7) representing the split extension possesses five maximal subgroups 2 3 : 7: 3, 2 3 : 4 , 4: 4 : 2 and two non-conjugate Klein's group 2 (7). The character tables of the groups and their maximal subgroups, tensor products and decompositions of their irreducible representations under the relevant maximal subgroups are identified. Possible implications in physics are discussed.

Sultan Qaboos University Journal for Science [SQUJS], 2016

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pse... more The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.

Nuclear Physics B, 1998

We report results of inclusive measurements of Λ • s, produced in the forward direction at the Sp... more We report results of inclusive measurements of Λ • s, produced in the forward direction at the SppS with √ s = 630 GeV, using the UA8 small angle Roman Pot spectrometers. These measurements cover the range in Feynman-x F and transverse momentum, 0.6 < x F < 1.0 and 0.4 < p t < 0.7 GeV, respectively. Within a systematic uncertainty of ±20% on the absolute cross section measurements, the results are indistinguishable from those made by some of us in two earlier experiments at the CERN Intersecting Storage Rings, with energies, √ s = 53 and 62 GeV. In the x F-range, 0.6-0.9, the absolute cross sections are lower by a factor of 2 to 3 than the predictions of the Lund model as embodied in the PYTHIA 5.6 and JETSET 7.3 Monte Carlo programs, indicating inadequacies in knowledge of the baryon fragmentation function. For the largest x F-range, 0.9-1.0, the measurements agree with the Monte Carlo predictions. We have measured the average Λ • polarization for our events and find (6 ± 12%), consistent with previous measurements at the ISR in the present region of x F-p t .

We report results of inclusive measurements of Lambdaoverlineos, produced in the forward directio... more We report results of inclusive measurements of Lambdaoverlineos, produced in the forward direction at the Sp p¯S with sqrt(s) = 630 GeV, using the UA8 small angle Roman Pot spectrometers. These measurements cover the range in Feynman-xF and transverse momentum, 0.6 < xF< 1.0 and 0.4 < pt< 0.7 GeV, respectively. Within a systematic uncertainty of +/-20% on the absolute cross section measurements, the results are indistinguishable from those made by some of us in two earlier experiments at the CERN Intersecting Storage Rings, with energies, sqrt(s) = 53 and 62 GeV. In the xF-range, 0.6-0.9, the absolute cross sections are lower by a factor of 2 to 3 than the predictions of the Lund model as embodied in the PYTHIA 5.6 and JETSET 7.3 Monte Carlo programs, indicating inadequacies in knowledge of the baryon fragmentation function. For the largest xF-range, 0.9-1.0, the measurements agree with the Monte Carlo predictions. We have measured the average Lambdaoverlineo polarizatio...

International Journal of Geometric Methods in Modern Physics, Apr 1, 2014

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional ... more 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 2 () Ih where h is the Coxeter number of the Coxeter group () WGwhich embeds the dihedral group h D of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A 4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine 4 () a WAcan be decomposed into two orthogonal spaces whose point groups is the dihedral group 5 D 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

arXiv (Cornell University), Feb 17, 2023

Quasicrystals described as the projections of higher dimensional cubic lattices, and the particul... more Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extension (2 (ℎ)) of the dihedral group (2 (ℎ)) of order 2h, ℎ = 2 being the Coxeter number, as a subgroup of affine group () offers a different perspective to h-fold symmetric quasicrystallography. The affine group (2 (ℎ)) is constructed as the subgroup of the affine group (), the symmetry of the cubic lattice ℤ. 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 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 copies of the h-gons tiled with some rhombi and rotated by the angle 2 ℎ. After a general discussion on the lattice ℤ with its affine group () embedding the affine dihedral group (2 (ℎ)) as a subgroup, its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry (), (= 1,2,3,4,5) have been presented and shown that the projection of the lattice ℤ 3 leads to the hexagonal lattice, the projection of the lattice ℤ 4 describes the Amman-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice ℤ 5 describes a quasicrystal structure with local 10fold 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.

arXiv (Cornell University), Nov 1, 2021

The projections of the lattices, may be used as models of quasicrystals, and the particular affin... more The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the 2 symmetry as a subgroup of 4 , discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine 2 is obtained as the subgroup of the affine 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell (0) of 4 project into the decagonal orbit of 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of 2. It is shown that the thick and thin rhombuses constitute the finite-fragments of the tiles of the Coxeter plane with the action of the affine 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 2 fixing a particular point on the Coxeter plane is determined

Acta Crystallographica, Apr 25, 2022

The projections of the lattices, may be used as models of quasicrystals, and the particular affin... more The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the 2 symmetry as a subgroup of 4 , discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine 2 is obtained as the subgroup of the affine 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell (0) of 4 project into the decagonal orbit of 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of 2. It is shown that the thick and thin rhombuses constitute the finite-fragments of the tiles of the Coxeter plane with the action of the affine 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 2 fixing a particular point on the Coxeter plane is determined

arXiv (Cornell University), Jun 8, 2016

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present t... more 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 subgroup of both the rank-4 Coxeter groups 44 4

Journal of physics, Nov 26, 2014

We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter... more We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter-Weyl group () an WB. An h-fold symmetric quasicrystal structure follows from the hyperqubic lattice whose point group is described by the Coxeter-Weyl group () n WB with the Coxeter number h=2n. Higher dimensional cubic lattices are explicitly constructed for 4,5,6 n  by identifying their rank-3 Coxeter subgroups and maximal dihedral subgroups. Decomposition of their Voronoi cells under the respective rank-3 subgroups 3 2 1 (), () () W A W H W A  and 3 () WH lead to the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice 4 B describes a quasicrystal structure with 8-fold symmetry. The 5 B lattice leads to quasicrystals with both 5fold and 10 fold symmetries. The lattice 6 B projects on a 12-fold symmetric quasicrystal as well as a 3D icosahedral quasicrystal depending on the choice of subspace of projections. The projected sets of lattice points are compatible with the available experimental data.

Journal of physics, Mar 1, 2011

4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl g... more 4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl groups. Their symmetries follow from the quaternionic descriptions of the rank-4 Coxeter-Dynkin diagrams. There exists a one to one correspondence between the finite subgroups of quaternions and the rank-4 Coxeter-Weyl groups. 1.

Journal of Physics: Conference Series

The present work offers a different perspective for the 5-fold symmetric quasicrystallography by ... more The present work offers a different perspective for the 5-fold symmetric quasicrystallography by employing affine H 2 as a subgroup of affine A 4. It is shown that the projection of the Voronoi cell of the root lattice A 4 can be dissociated as identical five decagons up to a rotation tiled by thick and thin rhombuses. 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 and two types of hexagons. Structure of the local dihedral symmetry H 2 fixing a particular point on the Coxeter plane is determined.

Acta Crystallographica Section A Foundations and Advances

The 3D facets of the Delone cells of the root latticeD6which tile the 6D Euclidean space in an al... more The 3D facets of the Delone cells of the root latticeD6which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 51/2)/2 with faces normal to the fivefold and threefold 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 from sixfundamentaltiles, the faces of which are normal to the fivefold 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 τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries...

Journal of Physics: Conference Series, 2014

One can obtain the quasicrystallographic structures from the projection of the higher dimensional... more One can obtain the quasicrystallographic structures from the projection of the higher dimensional lattices into 2D or 3D subspaces. Here we introduce a general technique applicable to any higher dimensional lattice described by the affine Coxeter groups. It is pointed out that the Coxeter number h and the Coxeter exponents play an important role in determining the principal planes onto which the lattice to be projected. The quasicrystal structures obtained by projection display the dihedral symmetry of order 2h . The projection subspaces are determined by using the eigenvectors and the corresponding eigenvalues of the Cartan matrix. Examples are given for 12-fold symmetric quasicrystal structures obtained by projections of the lattices determined by the affine Coxeter-Weyl groups Wa (F4), Wa (B6), and Wa (E6). The reflection generators R1 and R2 of the dihedral group D12 can be obtained as the products of the generators of the Coxeter-Weyl groups. It is noted that the quasicrystal structures obtained from the lattices Wa (F4) and Wa (B6) are compatible with the experimental data.

arXiv (Cornell University), Jun 8, 2016

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present t... more 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 subgroup of both the rank-4 Coxeter groups 44 4

Acta Crystallographica Section A Foundations and Advances, 2021

3D-facets of the Delone cells of the root lattice which tile the six-dimensional Euclidean space ... more 3D-facets of the Delone cells of the root lattice 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 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 generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and 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 with n th power of the golden ratio. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.

arXiv: Mathematical Physics, 2015

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) latti... more 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)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices have been constructed in terms of quaternions. Reflection planes of the Coxeter-Dynkin diagrams are identified with certain planes of the unit cube. It turns out that the pyritohedral symmetry takes a simpler form in terms of quaternionic representation. The D3 diagram is used to construct the vertices of polyhedra relevant to the cubic lattices and, in particular, constructions of the pseudoicosahedron and its dual pyritohedron are explicitly worked out.

SSRN Electronic Journal, 2019

Main motivation for this work is to evaluate students’ understanding of force and motion concepts... more Main motivation for this work is to evaluate students’ understanding of force and motion concepts in two calculus based introductory physics courses at Sultan Qaboos University, Muscat, Oman by using Force Concept inventory (FCI).

International Journal of Geometric Methods in Modern Physics, 2018

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see... more We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra projec...

Sultan Qaboos University Journal for Science [SQUJS], 2019

We analyze the group structures of two groups of order 1344 which are respectively non-split and ... more 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 2 (7). Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal. The group 2 3. 2 (7) is a finite subgroup of the Lie Group 2 preserving the set of octonions ± , (= 1,2, … ,7) representing a 7dimensional octahedron. Its three maximal subgroups 2 3 : 7: 3, 2 3. 4 and 4. 4 : 2 correspond to the finite subgroups of the Lie groups 2 , (4) and (3) respectively. The group 2 3 : 2 (7) representing the split extension possesses five maximal subgroups 2 3 : 7: 3, 2 3 : 4 , 4: 4 : 2 and two non-conjugate Klein's group 2 (7). The character tables of the groups and their maximal subgroups, tensor products and decompositions of their irreducible representations under the relevant maximal subgroups are identified. Possible implications in physics are discussed.

Sultan Qaboos University Journal for Science [SQUJS], 2016

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pse... more The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.

Nuclear Physics B, 1998

We report results of inclusive measurements of Λ • s, produced in the forward direction at the Sp... more We report results of inclusive measurements of Λ • s, produced in the forward direction at the SppS with √ s = 630 GeV, using the UA8 small angle Roman Pot spectrometers. These measurements cover the range in Feynman-x F and transverse momentum, 0.6 < x F < 1.0 and 0.4 < p t < 0.7 GeV, respectively. Within a systematic uncertainty of ±20% on the absolute cross section measurements, the results are indistinguishable from those made by some of us in two earlier experiments at the CERN Intersecting Storage Rings, with energies, √ s = 53 and 62 GeV. In the x F-range, 0.6-0.9, the absolute cross sections are lower by a factor of 2 to 3 than the predictions of the Lund model as embodied in the PYTHIA 5.6 and JETSET 7.3 Monte Carlo programs, indicating inadequacies in knowledge of the baryon fragmentation function. For the largest x F-range, 0.9-1.0, the measurements agree with the Monte Carlo predictions. We have measured the average Λ • polarization for our events and find (6 ± 12%), consistent with previous measurements at the ISR in the present region of x F-p t .

We report results of inclusive measurements of Lambdaoverlineos, produced in the forward directio... more We report results of inclusive measurements of Lambdaoverlineos, produced in the forward direction at the Sp p¯S with sqrt(s) = 630 GeV, using the UA8 small angle Roman Pot spectrometers. These measurements cover the range in Feynman-xF and transverse momentum, 0.6 < xF< 1.0 and 0.4 < pt< 0.7 GeV, respectively. Within a systematic uncertainty of +/-20% on the absolute cross section measurements, the results are indistinguishable from those made by some of us in two earlier experiments at the CERN Intersecting Storage Rings, with energies, sqrt(s) = 53 and 62 GeV. In the xF-range, 0.6-0.9, the absolute cross sections are lower by a factor of 2 to 3 than the predictions of the Lund model as embodied in the PYTHIA 5.6 and JETSET 7.3 Monte Carlo programs, indicating inadequacies in knowledge of the baryon fragmentation function. For the largest xF-range, 0.9-1.0, the measurements agree with the Monte Carlo predictions. We have measured the average Lambdaoverlineo polarizatio...