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1.
The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for \(i \in \mathbb {N}\). These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each \(i \in \mathbb {N}\). We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for \(i \le 4\), among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling. 相似文献
2.
Eric Rémila 《Discrete and Computational Geometry》2005,34(2):313-330
We fix two rectangles with integer dimensions. We give a quadratic
time algorithm which, given a polygon F as input, produces a tiling
of F with translated copies of our rectangles (or indicates that there is no
tiling). Moreover, we prove that any pair of tilings can be linked by a sequence of
local transformations of tilings, called flips. This study is based on the
use of Conway’s tiling groups and extends the results of Kenyon and Kenyon (limited to the
case when each rectangle has a side of length 1). 相似文献
3.
Thomas Stehling 《Discrete and Computational Geometry》1992,7(1):125-133
In 1988 Danzer [3] constructed a family of four tetrahedra which allows—with certain matching conditions—only aperiodic tilings. By analogy with the Ammann bars of planar Penrose tilings we define Ammann bars in space in the form of planar Penrose tilings we define Ammann bars in space in the form of plane sections of the four tetrahedra. If we require that the plane sections continue as planes across the faces of the tilings, we obtain an alternative matching condition, thus answering a question of Danzer. 相似文献
4.
LI JianLin College of Mathematics Information Science Shaanxi Normal University Xi'an China 《中国科学 数学(英文版)》2010,(5)
Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them. 相似文献
5.
《Discrete Mathematics》2022,345(3):112710
Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu. In this paper, we give a unified proof of these two shuffling theorems, which also covers the weighted case. Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp. Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem. Our proof also gives a simple explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings. 相似文献
6.
Frank 《Discrete and Computational Geometry》2008,29(3):459-467
Abstract. Tilings of R
2
can display hierarchy similar to that seen in the limit sequences of substitutions. Self-similarity for tilings has been
used as the standard generalization, but this viewpoint is limited because such tilings are analogous to limit points of constant-length
substitutions. To generalize limit points of non-constant-length substitutions, we define hierarchy for infinite, labelled
graphs, then extend this definition to tilings via their dual graphs. Examples of combinatorially substitutive tilings that
are not self-similar are given. We then find a sufficient condition for detecting combinatorial hierarchy that is motivated
by the characterization by Durand of substitutive sequences. That characterization relies upon the construction of the ``derived
sequence'—a recoding in terms of reappearances of an initial block. Following this, we define the ``derived Vorono? tiling'—a
retiling in terms of reappearances of an initial patch of tiles. Using derived Vorono? tilings, we obtain a sufficient condition
for a tiling to be combinatorially substitutive. 相似文献
7.
Natalie Priebe Frank 《Expositiones Mathematicae》2008,26(4):295-326
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions. 相似文献
8.
Richard L. Roth 《Geometriae Dedicata》1991,39(1):43-54
In this paper we describe and classify, using adjacency symbols, the 2-isohedral tilings of the plane such that all tiles have four edges and four tiles meet at each vertex. There are 69 such tilings. Since many of these can be constructed by dissecting isohedral tilings appropriately, we show which isohedral tilings are related in this way to these 2-isohedral tilings. 相似文献
9.
Thomas Kämpke 《Journal of Heuristics》1997,2(3):245-278
Labeling the vertices of a finite sequence of polygonal tilings with fewest monotonicity violations enables to represent the tilings by merely specifying sets of vertices—the sequences of their appearance results from the labels. Eventually, this allows a lossless data compression for the sequence of tilings.The existence and computation of suitable labelings is derived from matching and graph colorings which induce an order on the tilings. This order is series-parallel on each individual tiling. 相似文献
10.
11.
Generalizing results of Temperley (London Mathematical Society Lecture Notes Series 13 (1974) 202), Brooks et al. (Duke Math. J. 7 (1940) 312) and others (Electron. J. Combin. 7 (2000); Israel J. Math. 105 (1998) 61) we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to “discrete analytic functions” on the bipartite graph.The equivalence is extended to infinite periodic graphs, and we classify the resulting “almost periodic” tilings and harmonic functions. 相似文献
12.
We study the spaces of rhombus tilings, i.e. the
graphs whose vertices are tilings of a fixed zonotope. Two
tilings are linked if one can pass from one to the other
by a local
transformation, called a flip.
We first use a decomposition method to encode rhombus tilings and
give a useful characterization for a sequence of bits to encode a
tiling.
We use the previous coding to get a canonical
representation of tilings, and two order structures on the space of
tilings.
In codimension 2 we prove that the two order structures are equal.
In larger codimensions we study the lexicographic case, and get an order
regularity result. 相似文献
13.
Frank 《Discrete and Computational Geometry》2003,29(3):459-467
Abstract. Tilings of R
2
can display hierarchy similar to that seen in the limit sequences of substitutions. Self-similarity for tilings has been
used as the standard generalization, but this viewpoint is limited because such tilings are analogous to limit points of constant-length
substitutions. To generalize limit points of non-constant-length substitutions, we define hierarchy for infinite, labelled
graphs, then extend this definition to tilings via their dual graphs. Examples of combinatorially substitutive tilings that
are not self-similar are given. We then find a sufficient condition for detecting combinatorial hierarchy that is motivated
by the characterization by Durand of substitutive sequences. That characterization relies upon the construction of the ``derived
sequence'—a recoding in terms of reappearances of an initial block. Following this, we define the ``derived Vorono? tiling'—a
retiling in terms of reappearances of an initial patch of tiles. Using derived Vorono? tilings, we obtain a sufficient condition
for a tiling to be combinatorially substitutive. 相似文献
14.
Bridget Eileen Tenner 《Graphs and Combinatorics》2009,25(4):625-638
The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the
number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings,
modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k × 2k rectangle is congruent to 1 mod 4. 相似文献
15.
Noam Elkies Greg Kuperberg Michael Larsen James Propp 《Journal of Algebraic Combinatorics》1992,1(2):111-132
We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2
n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond. 相似文献
16.
Richard L. Roth 《Geometriae Dedicata》1989,29(2):185-191
The isohedral tilings of a ribbon or infinite strip are classified. There are 24 of them, 5 of which must be realized as either marked tilings or tilings of a ribbon with edges which are not straight lines. 相似文献
17.
Nicolau C. Saldanha 《Journal of Algebraic Combinatorics》2002,16(2):195-207
Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK* (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles. 相似文献
18.
Bridget Eileen Tenner 《Annals of Combinatorics》2010,14(4):553-568
This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight
tiling are that the “tiles” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions,
or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight
tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of
the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further
analysis of spotlight tilings will continue to yield elegant results and enumerations. 相似文献
19.
We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies et al. (1992) [2], [3]). Moreover, we consider the connection between the generating function and MacMahon's q-enumeration of plane partitions fitting in a given box 相似文献
20.
Andre Henriques 《Advances in Mathematics》2010,223(3):1107-1136
We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon - any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n−1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities. 相似文献