TILINGS WITH THE NEIGHBORHOOD PROPERTY

The neighborhood N(T) of a tile T is the set of all tiles which meet T in at least one point. If for each tile T there is a different tile T1 such that N(T) N(T1) then we say the tiling has the neighborhood property (NEBP). Cm:inbaum and Shepard conjecture that it is impossible to have a monohedral tiling of the plane such that every tile T has two different tiles TI,T2 with N(T) N(T) N(T:z). If all tiles are convex we show this conjecture is true by characterizing the convex plane tilings with NEBP. More precisely we prove that a convex plane tiling with NEBP has only triangular tiles and each tile has a 3-valent vertex. Removing 3-valent vertices and the incident edges from such a tiling yields an edge-to-edge planar triangulation. Conversely, given any edge-to-edge planar triangulation followed by insertion of a vertex and three edges that triangulate each triangle yields a convex plane tiling with NEBP. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We briefly discuss tilings of R3 with NEBP and exhibit a monohedral tetrahedral tiling of R3 with NEBP

Tilings with the neighborhood property

The neighborhood N(T) of a tile T is the set of all tiles which meet T in at least one point. If for each tile T there is a different tile T1 such that N(T)=N(T1) then we say the tiling has the neighborhood property (NEBP). Grünbaum and Shepard conjecture that it is impossible to have a monohedral tiling of the plane such that every tile T has two different tiles T1, T2 with N(T)=N(T1)=N(T2). If all tiles are convex we show this conjecture is true by characterizing the convex plane tilings with NEBP. More precisely we prove that a convex plane tiling with NEBP has only triangular tiles and each tile has a 3-valent vertex. Removing 3-valent vertices and the incident edges from such a tiling yields an edge-to-edge planar triangulation. Conversely, given any edge-to-edge planar triangulation followed by insertion of a vertex and three edges that triangulate each triangle yields a convex plane tiling with NEBP. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We briefly discuss tilings of R3 with NEBP and exhibit a monohedral tetrahedral tiling of R3 with NEBP

A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n log2 n)-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

Configuration space partitioning in tilings of a bounded region of the plane

Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much of this interest has revolved around fundamental geometrical problems such as minimizing the variety of tile types to be used, and also around important applications in areas such as crystallography as well as others. All these applications are of course confined to finite spatial regions, but in many cases they refer back directly to progress in tiling the whole, unbounded plane. Tilings of bounded regions of the plane have also been actively studied, but in general the additional complications imposed by the boundary conditions tend to constrain progress to mostly indirect results, such as recurrence relations. Here we study the tiling of rectangular regions of the plane by rectangular tiles. The tile types we use are squares, dominoes, and straight tetraminoes. For this set of tile types, not even recurrence relations seem to be available. Our approach is to seek to characterize this complex system through some fundamental physical quantities. We do this on two parallel tracks, one analytical for what seems to be the most complex special case still amenable to such approach, the other based on the Wang-Landau method for state-density estimation. Given a simple energy function based solely on tile contacts, we have found either approach to lead to illuminating depictions of entropy, temperature, and above all partitions of the configuration space. The notion of a configuration, in this context, refers to how many tiles of each type are used. We have found that certain partitions help bind together different aspects of the system in question and conjecture that future applications will benefit from the possibilities they afford.Comment: This version includes minor fixes and a new table, and updates metadat

On the structure of Ammann A2 tilings

We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete and Computational Geometry 20 (1998) 265-279]. By the same techniques we show that Ammann A2 tilings are not robust in the sense of [B. Durand, A. Romashchenko, A. Shen. Fixed-point tile sets and their applications, Journal of Computer and System Sciences, 78:3 (2012) 731--764]