Rhombille tiling

About: Rhombille tiling is a research topic. Over the lifetime, 329 publications have been published within this topic receiving 5039 citations.

A variational principle for domino tilings

Abstract: 1.1. Description of results. A domino is a 1 x 2 (or 2 x 1) rectangle, and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps. In 1961, Kasteleyn [Kal] found a formula for the number of domino tilings of an m x n rectangle (with mn even), as shown in Figure 1 for rm = n = 68. Temperley and Fisher [TF] used a differenit method and arrived at the same result at almost exactly the same time. Both lines of calculation showed that the logarithm of number of tilings, divided by the number of dominos in a tiling (that is, mn/2), converges to 2G/7r 0.58 (here G is Catalan's constant). On the other hand, in 1992 Elkies et al. [EKLP] studied domino tilings of regions they called Aztec diamonds (Figure 2 shows an Aztec diamond of order 48), and showed that the logarithm of the number of tilings, divided by the number of dominos, converges to the smaller number (log 2)/2 0.35. Thus, even though the region in Figure 1 has slightly smaller area than the region in Figure 2, the former has far more domino tilings. For regions with other shapes, neither of these asymptotic formulas may apply. In the present paper we consider simply-connected regions of arbitrary shape. We give an exact formula for the limiting value of the logarithm of the number of tilings per unit area, as a function of the shape of the boundary of the region, as the size of the region goes to infinity. In particular, we show that computation of this limit is intimately linked with an understanding of long-range variations in the local statistics of random domino tilings. Such variations can be seen by comparing Figures 1 and 2. Each of the two tilings is random in the sense that the algorithm [PWI that was used to create it generates each of the possible tilings of the region being tiled with the same probability. Hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

Loop tiling for parallelism

Abstract: List of Figures. List of Tables. Preface. Acknowledgments. Part I: Mathematic Background and Loop Transformation. 1. Mathematical Background. 2. Nonsingular Transformations and Permutability. Part II: Tiling as a Loop Transformation. 3. Rectangular Tiling. 4. Parallelepiped Tiling. Part III: Tiling for Distributed-Memory Machines. 5. SPMD Code Generation. 6. Communication-Minimal Tiling. 7. Time-Minimal Tiling. Bibliography. Index.

Random Tiling and Topological Defects in a Two-Dimensional Molecular Network

Abstract: A molecular network that exhibits critical correlations in the spatial order that is characteristic of a random, entropically stabilized, rhombus tiling is described. Specifically, we report a random tiling formed in a two-dimensional molecular network of p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite. The network is stabilized by hexagonal junctions of three, four, five, or six molecules and may be mapped onto a rhombus tiling in which an ordered array of vertices is embedded within a nonperiodic framework with spatial fluctuations in a local order characteristic of an entropically stabilized phase. We identified a topological defect that can propagate through the network, giving rise to a local reordering of molecular tiles and thus to transitions between quasi-degenerate local minima of a complex energy landscape. We draw parallels between the molecular tiling and dynamically arrested systems, such as glasses.

Alternating-Sign Matrices and Domino Tilings (Part II)

Abstract: We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.