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.Comment: 22 pages, 17 figure

Belt distance between facets of space-filling zonotopes

For every d-dimensional polytope P with centrally symmetric facets we can associate a "subway map" such that every line of this "subway" corresponds to set of facets parallel to one of ridges P. The belt diameter of P is the maximal number of line changes that you need to do in order to get from one station to another. In this paper we prove that belt diameter of d-dimensional space-filling zonotope is not greater than $\lceil \log_2\frac45d\rceil$. Moreover we show that this bound can not be improved in dimensions d at most 6.Comment: 17 pages, 5 figure

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.Comment: 26 pages, 17 figure

A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure