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Tiling with Polyominoes and Combinatorial Group Theory
 JOURNAL OF COMBINATORIAL THEORY, SERIES A 53, 1833208
, 1990
"... When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w ’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinat ..."
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When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w ’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinatorial grouptheoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled. Boundary invariants are used to solve problems concerning the tiling of triangularshaped regions of hexagons in the hexagonal lattice with certain tiles consisting of three hexagons. Boundary invariants give stronger conditions for nonexistence of tilings than those obtainable by weighting or coloring arguments. This is shown by considering whether or not a region has a signed tiling, which is a placement of tiles assigned weights 1 orI, such that all cells in the region are covered with total weight 1 and all cells outside with total weight 0. Any coloring (or weighting) argument that proves nonexistence of a tiling of a region also proves nonexistence of any signed tiling of the region as well. A partial converse holds: if a simply connected region has no signed tiling by simply connected tiles, then there is a generalized coloring argument proving that no signed tiling exists. There exist regions possessing a signed tiling which can be shown to have no perfect tiling using boundary invariants.
Leeuwen, Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino
 Inform. and Control
, 1984
"... Given N distinct memory modules, the elements of an (infinite) array in storage are distributed such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in ..."
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Given N distinct memory modules, the elements of an (infinite) array in storage are distributed such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector processing and SIMD machines. In 1975 Shapiro (IEEE Trans. Comput. C27 (1978), 42l~428) proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. A conjecture of Shapiro is settled and it is proved that for polyominos P a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme. (Periodicity implies a rapid technique to locate data elements.) The proof shows that when a polyomino P tessellates the plane without rotations or reflections, then it can tessellate the plane periodically, i.e., with the instances of P arranged in a lattice. It is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane, assuming the polyominos in the tessellation should all have an equal orientation. © 1984 Academic Press, Inc. 1.
A characterization of recognizable picture languages by tilings by finite sets
, 2002
"... As extension of the Kleene star to pictures, we introduce the operation of tiling. We give a characterization of recognizable picture languages by intersection of tilings by finite sets of pictures. ..."
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As extension of the Kleene star to pictures, we introduce the operation of tiling. We give a characterization of recognizable picture languages by intersection of tilings by finite sets of pictures.
Tiling Rectangles And Half Strips With Congruent Polyominoes
, 1996
"... In this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already kn ..."
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In this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already known to tile a rectangle, or were later found to tile a rectangle. It is still unknown if every polyomino that tiles a half strip also tiles a rectangle.
Tiling With Notched Cubes
, 1999
"... Golomb [1] showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger c ..."
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Golomb [1] showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a "notched cube") that tiles a larger copy of itself, but does not tile any box, and obtain several related results about tiling with this figure. We also obtain analogous results in all dimensions d 3.
KLARNER SYSTEMS AND TILING BOXES WITH POLYOMINOES
 JOURNAL OF COMBINATORIAL THEORY, SERIES A, 111 (2005), NO. 1, PP. 89–105.
, 2005
"... Let T be a protoset of ddimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by T? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of “prime ” boxes. All other ..."
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Let T be a protoset of ddimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by T? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of “prime ” boxes. All other boxes that can be tiled can be deduced from these prime boxes. We give a new, simpler proof of this fundamental result. We also show that there is no upper bound to the number of prime boxes, even when restricting attention to singleton protosets. In the last section, we determine the set of prime rectangles for several small polyominoes.
Tiling the Plane with a Fixed Number of polyominoes
, 2008
"... Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the ..."
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Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the problem is undecidable for sets of 11 polyominoes.
REFERENCES
"... our study. This illustrates the fallacy of extrapolating results obtained in patients with severe hypertension to patients with milder hypertension and emphasizes the necessity for carefully designed studies comparing side effect of drugs at equally effective doses in the same population. Acknowledg ..."
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our study. This illustrates the fallacy of extrapolating results obtained in patients with severe hypertension to patients with milder hypertension and emphasizes the necessity for carefully designed studies comparing side effect of drugs at equally effective doses in the same population. Acknowledgment The authors are grateful to Dr. Morton Leeds and the CIBAGeigy Corporation for the drugs, forms and support for this study. We wish to thank Dr. Dennis Gilliland for statistical advice and Mrs. Esther Stuart for secretarial assistance.
Enumeration of generalized polyominoes
"... As a generalization of polyominoes we consider edgetoedge connected nonoverlapping unions of regular kgons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular kgons. Additionally we give a table of the numbers ak(n) for small k and n obtained ..."
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As a generalization of polyominoes we consider edgetoedge connected nonoverlapping unions of regular kgons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular kgons. Additionally we give a table of the numbers ak(n) for small k and n obtained by computer enumeration. We finish with some open problems for kpolyominoes. 1
TILING WITH SIMILAR POLYOMINOES
 JOURNAL OF RECREATIONAL MATHEMATICS 31 (2002–2003), NO. 1, PP. 15–24.
, 2003
"... Numerous authors have studied tilings that use congruent copies of a single polyomino shape, for example ..."
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Numerous authors have studied tilings that use congruent copies of a single polyomino shape, for example