DATA STRUCTURES AND PROCEDURES FOR A POLYHEDRON ALGORITHM

In this paper we describe the data structures and the procedures of a program, which is based on the algorithms of [5,6]. Knowing the incidence structure of a polyhedron, the program finds all the essentially different facet pairings. The transformations, pairing the facets generate a space group, for which the polyhedron is a fundamental domain. The program also creates the defining relations of the group. Thus, we obtain discrete groups of certain combinatorial spaces. We have still to examine which groups can be realised in spaces of constant curvature (or in other simply connected spaces). Finally, we mention some results: Examining the 4-simplex, our program disproves Zhuk's conjecture concerning the number of essentially different facet pairings of d-simplices [11]. The classification of 3-simplex tilings has also been completed [7]. We have found the fundamental tilings of the Euclidean space with marked cubes and the corresponding crystalIographic groups [8]

Distances on Rhombus Tilings

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how "tight" rhombus tiling spaces are flip-connected. We introduce a lower bound (Hamming-distance) on the minimal number of flips to link two tilings (flip-distance), and we investigate whether it is sharp. The answer depends on the number n of different edge directions in the tiling: positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly negative for n=5 (decagonal tilings) or greater values of n. A standard proof is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led to a computer-assisted proof (whose main result can however be easily checked by hand).Comment: 18 pages, 9 figures, submitted to Theoretical Computer Science (special issue of DGCI'09

Maximal equicontinuous factors and cohomology for tiling spaces

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology