scholarly journals Explicit construction of the Voronoi and Delaunay cells ofW(An) andW(Dn) lattices and their facets

2018 ◽  
Vol 74 (5) ◽  
pp. 499-511 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Abeer Al-Siyabi ◽  
Ramazan Koc
Keyword(s):  
Voronoi Cell ◽  
Explicit Construction ◽  
Weyl Groups ◽  
Root Lattice ◽  
Coxeter Number ◽  
Aperiodic Tiling ◽  
Higher Dimensional ◽  
The Face ◽  
Coxeter Plane ◽  
Root Polytope

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter–Weyl groupsW(An) andW(Dn) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) 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 andn-hemicube and their volumes are calculated. The 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 centred at the origin of the root latticeAnare the polytopes of the fundamental weights ω1, ω2,…, ωnand the Delone cells of the root latticeDnare the polytopes obtained from the weights ω1, ωn−1and ωn. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n−1)-facet of the Voronoi cell of the root latticeAnis an (n−1)-dimensional rhombohedron and similarly the (n−1)-facet of the Voronoi cell of the root latticeDnis a dipyramid with a base of an (n−2)-cube. The volume of the Voronoi cell is calculatedviaits (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 latticesAn* andDn*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to theh-fold aperiodic tiling, wherehis the Coxeter number of the Coxeter–Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by theA4root lattice and the eightfold symmetric tessellation by theD5root lattice.

scholarly journals SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

2018 ◽  
Vol 15 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Abeer Al-Siyabi
Keyword(s):  
Dynkin Diagram ◽  
Voronoi Cell ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Unified Theory ◽  
Grand Unified Theory ◽  
Root Lattice ◽  
Gauge Bosons ◽  
Aperiodic Tiling ◽  
Coxeter Plane

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 project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to [Formula: see text] and even to [Formula: see text] by noting the Coxeter–Dynkin diagram embedding [Formula: see text]. Another embedding can be made through the relation [Formula: see text] for more popular [Formula: see text]. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups [Formula: see text] which can be obtained directly from [Formula: see text] by projection. This leads to relations of the [Formula: see text] polytopes with the quasicrystallography in 4D and [Formula: see text] polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group [Formula: see text].

scholarly journals Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesBn

2015 ◽  
Vol 71 (2) ◽  
pp. 175-185 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koc
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Icosahedral Symmetry ◽  
The Novel ◽  
Voronoi Cells ◽  
Aperiodic Tilings ◽  
Coxeter Number ◽  
Hypercubic Lattice ◽  
Higher Dimensional ◽  
Coxeter Plane

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter–Weyl groupWa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroupDhofW(Bn) withh= 2nrepresenting the Coxeter number describes theh-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed forn= 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroupsW(A3),W(H2) ×W(A1) andW(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the latticeB4onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. TheB5lattice is used to describe both fivefold and tenfold symmetries. The latticeB6can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry 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.

scholarly journals Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1082
Author(s):  
Ozdes Koca ◽  
Al-Siyabi ◽  
Koca ◽  
Koc
Keyword(s):  
General Scheme ◽  
Square Lattice ◽  
Face Centered Cubic ◽  
Root Lattice ◽  
Rhombic Dodecahedron ◽  
Orthogonal Projections ◽  
Seitz Cell ◽  
The Face ◽  
Face Centered ◽  
Coxeter Plane

The orthogonal projections of the Voronoi and Delone cells of root lattice An onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2h involving the Coxeter element of order h=n+1 of the Coxeter–Weyl group an play a crucial role for h-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4-fold symmetry.

scholarly journals Twelvefold symmetric quasicrystallography from the latticesF4,B6andE6

2014 ◽  
Vol 70 (6) ◽  
pp. 605-615 ◽  
Author(s):  
Nazife O. Koca ◽  
Mehmet Koca ◽  
Ramazan Koc
Keyword(s):  
Weyl Group ◽  
Coxeter Groups ◽  
Three Dimensional ◽  
Weyl Groups ◽  
Dimensional Euclidean Space ◽  
General Technique ◽  
Dimensional Lattice ◽  
Coxeter Number ◽  
Higher Dimensional ◽  
Group Play

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors inn-dimensional Euclidean space which lead to 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. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groupsWa(F4),Wa(B6) andWa(E6). These groups share the same Coxeter numberh= 12 with different Coxeter exponents. The dihedral subgroupD12of the Coxeter groups can be obtained by defining two generatorsR1andR2as the products of generators of the Coxeter–Weyl groups. The reflection generatorsR1andR2operate in the Coxeter planes where the Coxeter elementR1R2of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the latticesWa(F4) andWa(B6) are compatible with some experimental results.

scholarly journals EXPLICIT CONSTRUCTION OF COMPANION BASES

2015 ◽  
Vol 58 (2) ◽  
pp. 357-384
Author(s):  
MARK JAMES PARSONS
Keyword(s):  
Root System ◽  
Explicit Construction ◽  
Finitely Generated ◽  
Root Lattice ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Underlying Graph ◽  
Inner Products ◽  
Dynkin Quiver ◽  
Mutation Type

AbstractA companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a$\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in typeA(and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation typeAorD.

scholarly journals Beyond dimension reduction: Stable electric fields emerge from and allow representational drift

2021 ◽  
Author(s):  
Dimitris A. Pinotsis ◽  
Earl K. Miller
Keyword(s):  
Electric Fields ◽  
Neural Activity ◽  
Neural Ensemble ◽  
Higher Dimensional ◽  
Latent Space ◽  
The Face ◽  
Memory Content ◽  
The Stability ◽  
Dimensional Variable ◽  
The Brain

AbstractIt is known that the exact neurons maintaining a given memory (the neural ensemble) change from trial to trial. This raises the question of how the brain achieves stability in the face of this representational drift. Here, we demonstrate that this stability emerges at the level of the electric fields that arise from neural activity. The electric fields, in turn, can act as “guard rails” that funnel higher dimensional variable neural activity along stable lower dimensional routes. We show that electric fields carry information about working memory content. We obtained the latent space associated with each memory. We then confirmed the stability of the electric field by mapping the latent space to different cortical patches (that comprise a neural ensemble) and reconstructing information flow between patches. Stable electric fields can allow latent states to be transferred between brain areas, in accord with modern engram theory.

scholarly journals Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1018 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Weyl Groups ◽  
Root Lattice ◽  
Affine Weyl Groups ◽  
Fundamental Domains ◽  
Weyl Transforms ◽  
Orthogonality Relations ◽  
Discrete Transforms ◽  
Unitary Transform ◽  
Dual Affine ◽  
Root Lattices

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

scholarly journals FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions

2014 ◽  
Vol 19 (1) ◽  
pp. 95
Author(s):  
Nazife Özdeş Koca ◽  
Mehmet Koca ◽  
Muna Al-Sawafi
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Face Centered Cubic ◽  
Tetrahedral Symmetry ◽  
Orthogonal Projections ◽  
Cubic Lattices ◽  
Dynkin Diagrams ◽  
Octahedral Group ◽  
Face Centered ◽  
Coxeter Plane

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 extended Coxeter groups W(A3) and W(B3)=Aut(A3). It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices which have extensive applications in material science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices.   

scholarly journals Quasicrystal Tilings in Three Dimensions and Their Empires

Crystals ◽  
2018 ◽  
Vol 8 (10) ◽  
pp. 370 ◽  
Author(s):  
Dugan Hammock ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Projection Method ◽  
Voronoi Cell ◽  
Three Dimensions ◽  
New Method ◽  
Dimensional Lattice ◽  
Cell Complexes ◽  
Cubic Lattices ◽  
Higher Dimensional ◽  
Icosahedral Quasicrystals ◽  
Quasiperiodic Tilings

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

scholarly journals Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals

2014 ◽  
Vol 11 (04) ◽  
pp. 1450031 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koc
Keyword(s):  
Coxeter Group ◽  
Dihedral Group ◽  
Golden Ratio ◽  
Point Group ◽  
Canonical Projection ◽  
Face Centered Cubic ◽  
Coxeter Number ◽  
Decagonal Quasicrystals ◽  
Coxeter Plane ◽  
Group D

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 group 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.

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