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Trihexagonal tiling
About: Trihexagonal tiling is a research topic. Over the lifetime, 289 publications have been published within this topic receiving 4200 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
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
TL;DR: A molecular network that exhibits critical correlations in the spatial order that is characteristic of a random, entropically stabilized, rhombus tiling is described and a topological defect 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.
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.
206 citations
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01 Jan 2008TL;DR: The notion of tilings spaces and inverse limits was introduced in this article, where the authors propose a relaxation of the rules for tilings without finite local complexity, based on the notion of inverse limits.
Abstract: Basic notions Tiling spaces and inverse limits Cohomology of tilings spaces Relaxing the rules I: Rotations Pattern-equivariant cohomology Tricks of the trade Relaxing the rules II: Tilings without finite local complexity Solutions to selected exercises Bibliography.
164 citations
TL;DR: The Archimedean tiling (32.4.3.4) is a regular but complex polygonal assembly of equilateral triangles and squares as mentioned in this paper, which can lead to a new type of mesoscale self-organization.
Abstract: The Archimedean tiling (32.4.3.4) is a regular but complex polygonal assembly of equilateral triangles and squares. This tiling pattern with mesoscopic repeating distance has been found for an ABC star-branched three-component polymer composed of polyisoprene, polystyrene, and poly(2-vinylpyridine). In this structure the environment of a molecule splits into multiple sites and two microdomains with different sizes and shapes are formed for one component. This complexity is the first observation in complex polymer systems and can lead to a new type of mesoscale self-organization. The tiling pattern has been observed for the other materials on much shorter length-scale; therefore, the experimental fact observed in the present study is demonstrating that the complexity is universal over different hierarchies. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2427–2432, 2005
145 citations