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37
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 175 (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 34 (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 31 (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...
A strongly aperiodic set of tiles in the hyperbolic plane
 Inv. Math
"... Abstract We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a “regular production system ” [5] that does admit biinfinite orbi ..."
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Cited by 28 (3 self)
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Abstract We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a “regular production system ” [5] that does admit biinfinite orbits, but admits no periodic orbits. 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 extende ..."
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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...
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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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.
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 5 (3 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.
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 imm ..."
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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&quot;, developed further in [9]. 1.
Fixed point and aperiodic tilings
, 2008
"... An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile ..."
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Cited by 5 (3 self)
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An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene’s fixedpoint construction instead of geometric arguments. This construction is similar to J. von Neumann selfreproducing automata; similar ideas were also used by P. Gács in the context of errorcorrecting computations. The flexibility of this construction allows us to construct a “robust” aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. This property was not known for any of the existing aperiodic tile sets.
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 4 (3 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.