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# Hexagonal tiling

About: Hexagonal tiling is a research topic. Over the lifetime, 512 publications have been published within this topic receiving 6712 citations.

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TL;DR: In this article, the authors 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.

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

401 citations

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TL;DR: This work considers modelling and other applications, including the role of nearest neighbourhood in experimental design, the representation of connectivity in maps, and a new method for performing field surveys using hexagonal grids, which was demonstrated on montane heath vegetation.

385 citations

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TL;DR: This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

Abstract: This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

385 citations

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TL;DR: The honeycomb mesh, based on hexagonal plane tessellation, is considered as a multiprocessor interconnection network and honeycomb networks with rhombus and rectangle as the bounding polygons are considered.

Abstract: The honeycomb mesh, based on hexagonal plane tessellation, is considered as a multiprocessor interconnection network. A honeycomb mesh network with n nodes has degree 3 and diameter /spl ap/1.63/spl radic/n-1, which is 25 percent smaller degree and 18.5 percent smaller diameter than the mesh-connected computer with approximately the same number of nodes. Vertex and edge symmetric honeycomb torus network is obtained by adding wraparound edges to the honeycomb mesh. The network cost, defined as the product of degree and diameter, is better for honeycomb networks than for the two other families based on square (mesh-connected computers and tori) and triangular (hexagonal meshes and tori) tessellations. A convenient addressing scheme for nodes is introduced which provides simple computation of shortest paths and the diameter. Simple and optimal (in the number of required communication steps) routing, broadcasting, and semigroup computation algorithms are developed. The average distance in honeycomb torus with n nodes is proved to be approximately 0.54/spl radic/n. In addition to honeycomb meshes bounded by a regular hexagon, we consider also honeycomb networks with rhombus and rectangle as the bounding polygons.

300 citations

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TL;DR: The necessary conditions for the existence of such tilings using boundary invariants are given, which are combinatorial group-theoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled.

228 citations