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23
Wang Tiles for Image and Texture Generation
, 2003
"... We present a simple stochastic system for nonperiodically tiling the plane with a small set of Wang Tiles. The tiles may be filled with texture, patterns, or geometry that when assembled create a continuous representation. The primary advantage of using Wang Tiles is that once the tiles are filled, ..."
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Cited by 126 (4 self)
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We present a simple stochastic system for nonperiodically tiling the plane with a small set of Wang Tiles. The tiles may be filled with texture, patterns, or geometry that when assembled create a continuous representation. The primary advantage of using Wang Tiles is that once the tiles are filled, large expanses of nonperiodic texture (or patterns or geometry) can be created as needed very efficiently at runtime. Wang Tiles
A procedural object distribution function
 ACM TRANSACTIONS ON GRAPHICS
, 2005
"... In this paper, we present a procedural object distribution function, a new texture basis function that distributes procedurally generated objects over a procedurally generated texture. The objects are distributed uniformly over the texture, and are guaranteed not to overlap. The scale, size and orie ..."
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Cited by 24 (7 self)
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In this paper, we present a procedural object distribution function, a new texture basis function that distributes procedurally generated objects over a procedurally generated texture. The objects are distributed uniformly over the texture, and are guaranteed not to overlap. The scale, size and orientation of the objects can be easily manipulated. The texture basis function is efficient to evaluate, and is suited for realtime applications. The new texturing primitive we present extends the range of textures that can be generated procedurally. The procedural object distribution function we propose is based on Poisson disk tiles and a direct stochastic tiling algorithm for Wang tiles. Poisson disk tiles are square tiles filled with a precomputed set of Poisson disk distributed points, inspired by Wang tiles. A single set of Poisson disk tiles enables the realtime generation of an infinite amount of Poisson disk distributions of arbitrary size. With the direct stochastic tiling algorithm, these Poisson disk distributions can be evaluated locally, at any position in the Euclidean plane. Poisson disk tiles and the direct stochastic tiling algorithm have many other applications in computer graphics. We briefly explore applications in object distribution, primitive distribution for illustration, and environment map sampling.
An aperiodic set of 13 Wang tiles
"... A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a technique recently developed by J. Kari. The tilings simulate behavior of sequential machines that multiply real numbers in balanced representations by real constants. 1 Introduction Wang ti ..."
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Cited by 23 (0 self)
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A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a technique recently developed by J. Kari. The tilings simulate behavior of sequential machines that multiply real numbers in balanced representations by real constants. 1 Introduction Wang tiles are unit square tiles with colored edges. A tile set is a finite set of Wang tiles. We consider tilings of the infinite Euclidean plane using arbitrarily many copies of the tiles in the given tile set. The tiles are placed on the integer lattice points of the plane with their edges oriented horizontally and vertically. The tiles may not be rotated. A tiling is valid if everywhere the contiguous edges have the same color. Let T be a finite tile set, and f : ZZ 2 ! T a tiling. Tiling f is periodic with period (a; b) 2 ZZ 2 \Gamma f(0; 0)g iff f(x; y) = f(x + a; y + b) for every (x; y) 2 ZZ 2 . If there exists a periodic valid tiling with tiles of T , then there exists a doubly period...
Tiling groups for Wang tiles
 proceedings of the 13 th annual ACMSIAM Symposium On Discrete Algorithms (SODA) SIAM eds
"... We apply tiling groups and height functions to tilings of regions in the plane by Wang tiles, which are squares with colored boundaries where the colors of shared edges must match. We define a set of tiles as unambiguous if it contains all tiles equivalent to the identity in its tiling group. For al ..."
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Cited by 6 (3 self)
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We apply tiling groups and height functions to tilings of regions in the plane by Wang tiles, which are squares with colored boundaries where the colors of shared edges must match. We define a set of tiles as unambiguous if it contains all tiles equivalent to the identity in its tiling group. For all but one set of unambiguous tiles with two colors, we give efficient algorithms that tell whether a given region with colored boundary is tileable, show how to sample random tilings, and how to calculate the number of local moves or “flips ” required to transform one tiling into another. We also analyze the lattice structure of the set of tilings, and study several examples with three and four colors as well. 1
Global Cellular Automata
"... Global cellular automata are introduced as a generalization of 1dimensional cellular automata allowing the next state of a cell to depend on a "regular" global context rather than just a fixed size neighborhood. A number of well known results for 1dimensional cellular automata is extended to globa ..."
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Cited by 6 (0 self)
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Global cellular automata are introduced as a generalization of 1dimensional cellular automata allowing the next state of a cell to depend on a "regular" global context rather than just a fixed size neighborhood. A number of well known results for 1dimensional cellular automata is extended to global cellular automata. 1 Introduction Cellular automata (CA) are models of complex systems in which an infinite lattice of cells is updated in parallel according to a simple local rule. A dynamical system on the lattice of cells is a continuous and shiftinvariant function iff it can be specified by a CA. We will generalize 1dimensional CA to provide for a "regular" global context, while still using simple transition rules specified by a simple finite transducer called an !!sequential machine. Our global cellular automaton (GCA) will retain most of the properties of CA and at the same time allow us to define many noncontinuous transition functions. An important special case is the possibil...
Aperiodic Hierarchical Tilings
 in Foams, Emulsions, and Cellular Materials, Proceedings of NATOASI, E354
, 1999
"... Abstract. A substitution tiling is a certain globally de ned hierarchical structure in a geometric space. In [6] we show that for any substitution tiling in E n, n> 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immedi ..."
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Cited by 4 (1 self)
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Abstract. A substitution tiling is a certain globally de ned hierarchical structure in a geometric space. In [6] we show that for any substitution tiling in E n, n> 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. Here we give an expository account of the construction. In particular, we discuss the use of hierarchical, algorithmic, geometrically sensitive coordinates{ \addresses", developed further in [9]. 1.
Aperiodic sets of square tiles with colored corners
, 2006
"... In this paper we formalize the concept of square tiles with colored corners, a new kind of tiles closely related to Wang tiles. We construct aperiodic sets of square tiles with colored corners and several new aperiodic sets of Wang tiles using isomorphisms between tilings. The smallest aperiodic set ..."
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Cited by 3 (2 self)
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In this paper we formalize the concept of square tiles with colored corners, a new kind of tiles closely related to Wang tiles. We construct aperiodic sets of square tiles with colored corners and several new aperiodic sets of Wang tiles using isomorphisms between tilings. The smallest aperiodic set of square tiles with colored corners we have created consists of 44 tiles over 6 colors.
STRUCTURAL ASPECTS OF TILINGS
, 2008
"... In this paper, we study the structure of the set of tilings produced by any given tileset. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in two different contexts: the first one is combinatorial and th ..."
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Cited by 3 (2 self)
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In this paper, we study the structure of the set of tilings produced by any given tileset. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in two different contexts: the first one is combinatorial and the other topological. These two approaches have independent merits and, once combined, provide somehow surprising results. The particular case where the set of produced tilings is countable is deeply investigated while we prove that the uncountable case may have a completely different structure. We introduce a pattern preorder and also make use of CantorBendixson rank. Our first main result is that a tileset that produces only periodic tilings produces only a finite number of them. Our second main result exhibits a tiling with exactly one vector of periodicity in the countable case.
A Small Aperiodic Set of Tiles
"... We give a simple set of two tiles that can only tile aperiodically  that is no tiling with these tiles is invariant under any in nite cyclic group of isometries. Although general constructions for producing aperiodic sets of tiles are nally appearing, simple aperiodic sets are fairly rare. This set ..."
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Cited by 2 (0 self)
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We give a simple set of two tiles that can only tile aperiodically  that is no tiling with these tiles is invariant under any in nite cyclic group of isometries. Although general constructions for producing aperiodic sets of tiles are nally appearing, simple aperiodic sets are fairly rare. This set is among the smallest sets ever found. A tiling is nonperiodic if there is no in nite cyclic group of isometries leaving the tiling invariant. In E 2, this is equivalent to requiring that no translation leaves the tiling invariant. A set of tiles is aperiodic if it is possible to completely tile the plane with comgruent copies of the tiles, but only nonperiodically. For example, a pair of unit squares, one black and one white, is not an aperiodic set of tiles: it is possible to tile nonperiodically with black and white squares but they can tile periodically as well. Here we give a new, simple example of a set of aperiodic tiles, the T (trilobite) and C (cross) ( gure 1); in any tiling with these tiles, we will require that the \tips " of the tiles meet as pictured at right. (A local condition such as this is a \matching rule"). Two variations of the tiles are given at the end of this paper. These tiles are among the simplest ever found, and are related to a a family of aperiodic sets of 2 tiles in each E n, n 3 [10]. The reader may wish to examine a photocopy of the appendix with a pair of scissors. trilobite cross tips must meet like so: Figure 1: The Trilobite and Cross It has been many years since an aperiodic set of, say, fewer than ve tiles has been found. In all, this new set is only one of a handful of known aperiodic
High Complexity Tilings with Sparse Errors
, 2009
"... Tile sets and tilings of the plane appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). The idea is to enforce some global properties (of the entire tiling) by means of local rules (for neighbor tiles). A fundamental question: Can local rules enforce a comp ..."
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Cited by 2 (1 self)
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Tile sets and tilings of the plane appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). The idea is to enforce some global properties (of the entire tiling) by means of local rules (for neighbor tiles). A fundamental question: Can local rules enforce a complex (highly irregular) structure of a tiling? The minimal (and weak) notion of irregularity is aperiodicity. R. Berger constructed a tile set such that every tiling is aperiodic. Though Berger’s tilings are not periodic, they are very regular in an intuitive sense. In [3] a stronger result was proven: There exists a tile set such that all n×n squares in all tilings have Kolmogorov complexity Ω(n), i.e., contain Ω(n) bits of information. Such a tiling cannot be periodic or even computable. In the present paper we apply the fixedpoint argument from [5] to give a new construction of a tile set that enforces high Kolmogorov complexity tilings (thus providing an alternative proof of the results of [3]). The new construction is quite flexible, and we use it to prove a much stronger result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.