Journal articles: 'Tiling (mathematics)'

1

Fosnaugh, Linda S., and Marvin E. Harrell. "Covering the Plane with Rep-Tiles." Mathematics Teaching in the Middle School 1, no. 8 (January 1996): 666–70. http://dx.doi.org/10.5951/mtms.1.8.0666.

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Because of its beauty and applicability in ancient as well as modern times, the art of tiling has interested humankind for centuries. Roman mosaics and Moslem religious buildings are among some of the oldest examples of tiling. More modern examples of tiling as an art form can be found in the works of the Dutch artist M. C. Escher. Senechal (1990) notes that the study of tilings is important in mathematics education because the study of shape “draws on and contributes to not only mathematics but also the sciences and the arts.” Some common uses of tilings include the covering of walls, floors, ceilings, streets, sidewalks, and patios.

2

WANG, XINCHANG, PEICHANG OUYANG, KWOKWAI CHUNG, XIAOGEN ZHAN, HUA YI, and XIAOSONG TANG. "FRACTAL TILINGS FROM SUBSTITUTION TILINGS." Fractals 27, no. 02 (March 2019): 1950009. http://dx.doi.org/10.1142/s0218348x19500099.

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A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

3

KWAPISZ, JAROSLAW. "Rigidity and mapping class group for abstract tiling spaces." Ergodic Theory and Dynamical Systems 31, no. 6 (March 14, 2011): 1745–83. http://dx.doi.org/10.1017/s0143385710000696.

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AbstractWe study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝdon a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.

4

Amano, Kazuyuki, and Yoshinobu Haruyama. "On the Number of p4-Tilings by an n-Omino." International Journal of Computational Geometry & Applications 29, no. 01 (March 2019): 3–19. http://dx.doi.org/10.1142/s0218195919400016.

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A plane tiling by the copies of a polyomino is called isohedral if every pair of copies in the tiling has a symmetry of the tiling that maps one copy to the other. We show that, for every [Formula: see text]-omino (i.e., polyomino consisting of [Formula: see text] cells), the number of non-equivalent isohedral tilings generated by 90 degree rotations, so called p4-tilings or quarter-turn tilings, is bounded by a constant (independent of [Formula: see text]). The proof relies on the analysis of the factorization of the boundary word of a polyomino. We also show an example of a polyomino that has three non-equivalent p4-tilings.

5

AVELINO, CATARINA P., and ALTINO F. SANTOS. "DEFORMATION OF F-TILINGSVERSUSDEFORMATION OF ISOMETRIC FOLDINGS." International Journal of Mathematics 23, no. 09 (July 31, 2012): 1250092. http://dx.doi.org/10.1142/s0129167x12500929.

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We present some relations between deformation of spherical isometric foldings and deformation of spherical f-tilings. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. It is conjectured that any f-tiling is (continuously) deformable in the standard f-tiling τs= {(x, y, z) ∈ S2: z = 0} and it is shown that the deformation of f-tilings does not induce a continuous deformation on its associated isometric foldings.

6

DOMBI, E. R., and N. D. GILBERT. "THE TILING SEMIGROUPS OF ONE-DIMENSIONAL PERIODIC TILINGS." Journal of the Australian Mathematical Society 87, no. 2 (July 23, 2009): 153–60. http://dx.doi.org/10.1017/s144678870800075x.

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AbstractA one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word.

7

Kurz, Terri L. "The Mathematics of Tiling." Teaching Children Mathematics 20, no. 7 (March 2014): 416–17. http://dx.doi.org/10.5951/teacchilmath.20.7.0416.

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People who lay tile for a living use mathematics every day to decide how much tile, grout, and other supplies are required to complete each job. Measurement and geometry are an integral part of designing tile patterns. Collections of short activities focus on a monthly theme that includes four activities each for grade bands K–2, 3–4, and 5–6 and aims for an inquiry or problem-solving orientation.

8

Lidjan, Edin, and Ðordje Baralic. "Homology of polyomino tilings on flat surfaces." Applicable Analysis and Discrete Mathematics, no. 00 (2021): 31. http://dx.doi.org/10.2298/aadm210307031l.

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The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a ?coloring proof? of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.

9

Xu, You. "Fractal n-hedral tilings of ℝd." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 66, no. 3 (June 1999): 403–17. http://dx.doi.org/10.1017/s1446788700036697.

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AbstractAn n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

10

CANNON, S., D. A. LEVIN, and A. STAUFFER. "Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings." Combinatorics, Probability and Computing 28, no. 3 (October 31, 2018): 365–87. http://dx.doi.org/10.1017/s0963548318000470.

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We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

11

Wheatley, Grayson H. "Research into Practice: Spatial Sense and the Construction of Abstract Units in Tiling." Arithmetic Teacher 39, no. 8 (April 1992): 43–45. http://dx.doi.org/10.5951/at.39.8.0043.

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Although tiling has been a supplementary topic in elementary school mathematics for many years, its value has not been fully appreciated. This article dicusses a variation on tiling that offers rich opportunities for the construction of fundamental mathematical relationships. In particular, unitizing as a mental operation is identified as a basis for much mathematical activity in both geometric and numerical settings.

12

Muzika-Dizdarevic, Manuela, and Rade Zivaljevic. "Symmetric polyomino tilings, tribones, ideals, and Gröbner bases." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 1–23. http://dx.doi.org/10.2298/pim1512001m.

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We apply the theory of Grobner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions TN = T3k?1 and TN = T3k in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120? rotation of the triangle if and only if either N = 27r ? 1 or N = 27r for some integer r > 0. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.

13

JULIEN, ANTOINE. "Complexity and cohomology for cut-and-projection tilings." Ergodic Theory and Dynamical Systems 30, no. 2 (June 23, 2009): 489–523. http://dx.doi.org/10.1017/s0143385709000194.

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AbstractWe consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponentαin terms of the ranks of certain groups which appear in the construction. We give bounds forα. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

14

BALOGH, JÓZSEF, ANDREW TREGLOWN, and ADAM ZSOLT WAGNER. "Tilings in Randomly Perturbed Dense Graphs." Combinatorics, Probability and Computing 28, no. 2 (July 16, 2018): 159–76. http://dx.doi.org/10.1017/s0963548318000366.

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A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

15

Tenner, Bridget Eileen. "Spotlight Tiling." Annals of Combinatorics 14, no. 4 (December 2010): 553–68. http://dx.doi.org/10.1007/s00026-011-0077-6.

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16

Yang, Qi, and Chuanming Zong. "Multiple Lattice Tilings in Euclidean Spaces." Canadian Mathematical Bulletin 62, no. 4 (November 16, 2018): 923–29. http://dx.doi.org/10.4153/s0008439518000103.

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AbstractIn 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

17

Narayan, Darren A., and Allen J. Schwenk. "Tiling Large Rectangles." Mathematics Magazine 75, no. 5 (December 1, 2002): 372. http://dx.doi.org/10.2307/3219068.

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18

Mann, Casey. "Heesch's Tiling Problem." American Mathematical Monthly 111, no. 6 (June 2004): 509. http://dx.doi.org/10.2307/4145069.

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19

Thurston, William P. "Conway's Tiling Groups." American Mathematical Monthly 97, no. 8 (October 1990): 757–73. http://dx.doi.org/10.1080/00029890.1990.11995660.

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20

Komlós, János. "Tiling Turán Theorems." Combinatorica 20, no. 2 (February 1, 2000): 203–18. http://dx.doi.org/10.1007/s004930070020.

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21

PALAGALLO, JUDITH, and MARIA SALCEDO. "SYMMETRIES OF FRACTAL TILINGS." Fractals 16, no. 01 (March 2008): 69–78. http://dx.doi.org/10.1142/s0218348x08003806.

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Self-similar tilings of the plane can be generated by an iterative replacement property using squares and triangles. We show tilings with tiles that are topological disks and others whose tiles are connected but have an interior that is disconnected. In each example we note the symmetries of the individual tiles and describe how each tiling exhibits a property common to crystallographic tilings.

22

BARGE, MARCY, BEVERLY DIAMOND, JOHN HUNTON, and LORENZO SADUN. "Cohomology of substitution tiling spaces." Ergodic Theory and Dynamical Systems 30, no. 6 (November 4, 2009): 1607–27. http://dx.doi.org/10.1017/s0143385709000777.

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AbstractAnderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

23

Mendelsohn, N. S. "Tiling with Dominoes." College Mathematics Journal 35, no. 2 (March 2004): 115. http://dx.doi.org/10.2307/4146865.

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Mendelsohn, N. S. "Tiling with Dominoes." College Mathematics Journal 35, no. 2 (March 2004): 115–20. http://dx.doi.org/10.1080/07468342.2004.11922062.

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25

Vassallo, Salvatore. "BUFFON’S COIN AND NEEDLE PROBLEMS FOR THE SNUB HEXAGONAL TILING." Advances in Mathematics: Scientific Journal 10, no. 4 (April 29, 2021): 2223–33. http://dx.doi.org/10.37418/amsj.10.4.36.

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26

Vassallo, Salvatore. "BUFFON’S COIN AND NEEDLE PROBLEMS FOR THE RHOMBITRIHEXAGONAL TILING." Advances in Mathematics: Scientific Journal 11, no. 3 (March 22, 2022): 197–209. http://dx.doi.org/10.37418/amsj.11.3.5.

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In this paper we consider the rhombitrihexagonal tiling of the plane $(3, 4, 6, 4)$ Archimedean tiling and compute the probability that a random circle or a random segment intersects a side of the tiling.

27

LAFITTE, GREGORY, and MICHAEL WEISS. "Tilings: simulation and universality." Mathematical Structures in Computer Science 20, no. 5 (October 2010): 813–50. http://dx.doi.org/10.1017/s0960129510000228.

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Wang tiles are unit-size squares with coloured edges. In this paper, we approach one aspect of the study of tiling computability: the quest for a universal tile set. Using a complex construction, based on Robinson's classical construction and its different modifications, we build a tile set (pronounced ayin) that almost always simulates any tile set. By way of Banach–Mazur games on tilings topological spaces, we prove that the set of -tilings that do not satisfy the universality condition is meagre in the set of -tilings.

28

Vince, Andrew. "Rep-tiling Euclidean space." Aequationes Mathematicae 50, no. 1-2 (August 1995): 191–213. http://dx.doi.org/10.1007/bf01831118.

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29

Gage, Jenny. "Reality maths: mathematics lessons from regular floor tiling." Journal of Mathematics and the Arts 3, no. 3 (September 2009): 135–42. http://dx.doi.org/10.1080/17513470903149945.

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30

Stenger, Florian, and Axel Voigt. "Towards Infinite Tilings with Symmetric Boundaries." Symmetry 11, no. 4 (March 27, 2019): 444. http://dx.doi.org/10.3390/sym11040444.

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Large-time coarsening and the associated scaling and statistically self-similar properties are used to construct infinite tilings. This is realized using a Cahn–Hilliard equation and special boundaries on each tile. Within a compromise between computational effort and the goal to reduce recurrences, an infinite tiling has been created and software which zooms in and out evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. We also analyze the scaling behavior and the statistically self-similar properties and describe the numerical approach, which is based on finite elements and an energy-stable time discretization.

31

Jepsen, Charles H. "Tiling with Incomparable Cuboids." Mathematics Magazine 59, no. 5 (December 1, 1986): 283. http://dx.doi.org/10.2307/2689403.

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32

Metrebian, Harry. "TILING WITH PUNCTURED INTERVALS." Mathematika 65, no. 2 (October 29, 2018): 181–89. http://dx.doi.org/10.1112/s0025579318000384.

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33

GABARDO, JEAN-PIERRE, and XIAOJIANG YU. "NATURAL TILING, LATTICE TILING AND LEBESGUE MEASURE OF INTEGRAL SELF-AFFINE TILES." Journal of the London Mathematical Society 74, no. 01 (August 2006): 184–204. http://dx.doi.org/10.1112/s0024610706022915.

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34

BRAIDES, ANDREA, and MARGHERITA SOLCI. "INTERFACIAL ENERGIES ON PENROSE LATTICES." Mathematical Models and Methods in Applied Sciences 21, no. 05 (May 2011): 1193–210. http://dx.doi.org/10.1142/s0218202511005295.

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In this paper we prove a homogenization theorem for interfacial discrete energies defined on an a-periodic Penrose tiling in ℝ2. A general result on the homogenization of surface energies cannot be directly adapted to this case; the existence of the limit interfacial energy is therefore proved by showing some refined "quasi-periodic" properties of the tilings.

35

Chlebus, Bogdan S. "Domino-tiling games." Journal of Computer and System Sciences 32, no. 3 (June 1986): 374–92. http://dx.doi.org/10.1016/0022-0000(86)90036-x.

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36

Paris-Romaskevich, O. "Tiling billiards and Dynnikov’s helicoid." Transactions of the Moscow Mathematical Society 82 (March 15, 2022): 133–47. http://dx.doi.org/10.1090/mosc/317.

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Here are two problems. First, understanding the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describing the topology of connected components of plane sections of a centrally symmetric subsurface S ⊂ T 3 S \subset \mathbb {T}^3 of genus 3 3 . In this paper we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov’s problem in higher genus seems promising, as we show at the end of this paper.

37

GARVER, ALEXANDER, and THOMAS MCCONVILLE. "ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS." Glasgow Mathematical Journal 62, no. 1 (February 7, 2019): 147–82. http://dx.doi.org/10.1017/s0017089519000028.

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AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

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Kolountzakis, Mihail N., and Máté Matolcsi. "Algorithms for translational tiling." Journal of Mathematics and Music 3, no. 2 (July 2009): 85–97. http://dx.doi.org/10.1080/17459730903040899.

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39

Holladay, Kenneth. "Estimating the Size of Context-Free Tiling Languages." Canadian Journal of Mathematics 39, no. 6 (December 1, 1987): 1413–33. http://dx.doi.org/10.4153/cjm-1987-066-4.

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The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.

40

Nagai, Yasushi. "Absence of absolutely continuous diffraction spectrum for certain S-adic tilings." Nonlinearity 34, no. 11 (October 21, 2021): 7963–90. http://dx.doi.org/10.1088/1361-6544/ac2a51.

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Abstract Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R d , which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for ‘almost all’ binary block-substitution cases and thus prove the absence of the absolutely continuous diffraction spectrum for most of S-adic tilings from a family of binary block substitutions.

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MALONEY, GREGORY R., and DAN RUST. "Beyond primitivity for one-dimensional substitution subshifts and tiling spaces." Ergodic Theory and Dynamical Systems 38, no. 3 (September 20, 2016): 1086–117. http://dx.doi.org/10.1017/etds.2016.58.

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We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

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Ash, J. Marshall, and Solomon W. Golomb. "Tiling Deficient Rectangles with Trominoes." Mathematics Magazine 77, no. 1 (February 1, 2004): 46. http://dx.doi.org/10.2307/3219230.

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Griggs, Jerrold R., Michael Woltermann, B. Borchers, D. Callan, R. J. Chapman, J. Chilcott, P. Cull, et al. "Tiling Rectangles with Trominoes: 10641." American Mathematical Monthly 107, no. 2 (February 2000): 179. http://dx.doi.org/10.2307/2589452.

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MacKinnon, Nick, and John C. Cock. "Tiling Squares with Rectangles: 10883." American Mathematical Monthly 110, no. 4 (April 2003): 343. http://dx.doi.org/10.2307/3647894.

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45

Johnsonbaugh, Richard. "Tiling Deficient Boards with Trominoes." Mathematics Magazine 59, no. 1 (February 1, 1986): 34. http://dx.doi.org/10.2307/2690016.

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46

Protasov, V. Yu, and T. I. Zaitseva. "Self-Affine Tiling of Polyhedra." Doklady Mathematics 104, no. 2 (September 2021): 267–72. http://dx.doi.org/10.1134/s1064562421050112.

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47

Hall, Rachel W., and Paul Klingsberg. "Asymmetric Rhythms and Tiling Canons." American Mathematical Monthly 113, no. 10 (December 1, 2006): 887. http://dx.doi.org/10.2307/27642087.

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48

Kolountzakis, Mihail N., and Thomas Wolff. "On the Steinhaus tiling problem." Mathematika 46, no. 2 (December 1999): 253–80. http://dx.doi.org/10.1112/s0025579300007750.

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49

Walton, James J. "Cohomology of rotational tiling spaces." Bulletin of the London Mathematical Society 49, no. 6 (October 6, 2017): 1013–27. http://dx.doi.org/10.1112/blms.12098.

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50

Ash, J. Marshall, and Solomon W. Golomb. "Tiling Deficient Rectangles with Trominoes." Mathematics Magazine 77, no. 1 (February 2004): 46–55. http://dx.doi.org/10.1080/0025570x.2004.11953226.

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Journal articles on the topic 'Tiling (mathematics)'

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