Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

Four Dimensional Quantum Topology Changes of Spacetimes

We investigate topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant. As Riemannian manifolds which describe quantum tunnelings of spacetime we consider constant negative curvature solutions of the Einstein equation i.e. hyperbolic geometries. Using four dimensional polytopes, we can explicitly construct hyperbolic manifolds with topologically non-trivial boundaries which describe topology changes. These instanton-like solutions are constructed out of 8-cell's, 16-cell's or 24-cell's and have several points at infinity called cusps. The hyperbolic manifolds are non-compact because of the cusps but have finite volumes. Then we evaluate topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds. We find that the more complicated are the topology changes, the more likely are suppressed.Comment: 26 pages, revtex, 13 figures. The calculation of volume and grammatical errors are correcte

On the sum of the Voronoi polytope of a lattice with a zonotope

A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to be free. Given a set $U$ of free vectors, we give several methods for checking if $P + Z(U)$ is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. In the case of the root lattice $\mathsf{E}_6$, it is possible to give a more geometric description of the admissible sets of vectors $U$. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of $27$ lines in a cubic. Based on a detailed study of the geometry of $P_V(\mathsf{e}_6)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V(\mathsf{E}_6) + Z(U)$ is a parallelotope. The enumeration yields $10$ maximal families of vectors, which are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table

Stochastic model for the 3D microstructure of pristine and cyclically aged cathodes in Li-ion batteries

It is well-known that the microstructure of electrodes in lithium-ion batteries strongly affects their performance. Vice versa, the microstructure can exhibit strong changes during the usage of the battery due to aging effects. For a better understanding of these effects, mathematical analysis and modeling has turned out to be of great help. In particular, stochastic 3D microstructure models have proven to be a powerful and very flexible tool to generate various kinds of particle-based structures. Recently, such models have been proposed for the microstructure of anodes in lithium-ion energy and power cells. In the present paper, we describe a stochastic modeling approach for the 3D microstructure of cathodes in a lithium-ion energy cell, which differs significantly from the one observed in anodes. The model for the cathode data enhances the ideas of the anode models, which have been developed so far. It is calibrated using 3D tomographic image data from pristine as well as two aged cathodes. A validation based on morphological image characteristics shows that the model is able to realistically describe both, the microstructure of pristine and aged cathodes. Thus, we conclude that the model is suitable to generate virtual, but realistic microstructures of lithium-ion cathodes

The width of 5-dimensional prismatoids

Santos' construction of counter-examples to the Hirsch Conjecture (2012) is based on the existence of prismatoids of dimension d of width greater than d. Santos, Stephen and Thomas (2012) have shown that this cannot occur in $d \le 4$. Motivated by this we here study the width of 5-dimensional prismatoids, obtaining the following results: - There are 5-prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos' original construction. This leads to non-Hirsch polytopes of dimension 20, rather than the original dimension 43. - There are 5-prismatoids with $n$ vertices and width $\Omega(\sqrt{n})$ for arbitrarily large $n$. Hence, the width of 5-prismatoids is unbounded.Comment: 31 pages, 10 figures. Changes from v1: the introduction has been edited, and a minor correction made in the statement of Proposition 1.

Dynamical partitions of space in any dimension

Topologically stable cellular partitions of D dimensional spaces are studied. A complete statistical description of the average structural properties of such partition is given in term of a sequence of D/2-1 (or (D-1)/2) variables for D even (or odd). These variables are the average coordination numbers of the 2k-dimensional polytopes (2k < D) which make the cellular structure. A procedure to built D dimensional space partitions trough cell-division and cell-coalescence transformations is presented. Classes of structures which are invariant under these transformations are found and the average properties of such structures are illustrated. Homogeneous partitions are constructed and compared with the known structures obtained by Voronoi partitions and sphere packings in high dimensions.Comment: LaTeX 5 eps figures, submetted to J. Phys.