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1.
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
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.
N. C. Saldanha C. Tomei M. A. Casarin Jr. D. Romualdo 《Discrete and Computational Geometry》1995,14(1):207-233
We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares.
Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected
component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces,
the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected
components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein
bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these
geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected.
The first two authors received support from SCT and CNPq, Brazil. The other two were supported by a grant for undergraduates
by CNPq. 相似文献
4.
We produce an algorithm that is optimal with respect to both space and execution time to generate all the lozenge (or domino) tilings of a hole-free, general-shape domain given as input.We first recall some useful results, namely the distributive lattice structure of the space of tilings and Thurston's algorithm for constructing a particular tiling. We then describe our algorithm and study its complexity. 相似文献
5.
We describe the class of Archimedean polyhedra in the three-dimensional Lobachevsky space, which technically reduces to studying
Archimedean tilings of the Lobachevsky plane. We analyze the possibility of obtaining Archimedean tilings by methods that
are usually applied on the sphere and in the Euclidean plane. It is pointed out that such tilings can be constructed by using
certain types of Fedorov groups in the Lobachevsky plane. We propose a general approach to the problem of classifying Archimedean
tilings of the Lobachevsky plane. 相似文献
6.
7.
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
8.
Prof. Dr. Lothar Collatz 《Journal of Geometry》1988,31(1-2):42-64
We consider double-periodic tilings of the whole plane from the view of graph theory, not with respect to symmetry groups. We suppose that the graph is planar and connected and that the fundamental domain contains a finite number of vertices and edges. We assign to every tiling a tableau. There exists a fundamental formula connecting the number of all numbers of the tableau with the sum of the reciprocals of all these numbers and with the number p of lines in the tableau; the formula is proved even if multiple edges or loops occur. By this way we get a graph-theoretic classification of the tilings. We introduce families F of tilings and their ranks. The family F={k1,k2...,ks} (with k1>k2>...>ks>O) is the set of all tilings, the tableau of which contains all the numbers kj and no others. The smallest number p of lines (which occur for the tilings of the family F) is the rank of F and has special geometric interest. Some open questions are mentioned at the end.
Herrn Helmut Karzel zum 60. Geburtstag gewidmet 相似文献
Herrn Helmut Karzel zum 60. Geburtstag gewidmet 相似文献
9.
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems. 相似文献
10.
Penrose tilings as coverings of congruent decagons 总被引:1,自引:0,他引:1
Petra Gummelt 《Geometriae Dedicata》1996,62(1):1-17
The open problem of tiling theory whether there is a single aperiodic two-dimensional prototile with corresponding matching rules, is answered for coverings instead of tilings. We introduce admissible overlaps for the regular decagon determining only nonperiodic coverings of the Euclidean plane which are equivalent to tilings by Robinson triangles. Our work is motivated by structural properties of quasicrystals. 相似文献
11.
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
12.
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
13.
We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well
known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution
is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures
are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some
mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of
non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings,
the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling. 相似文献
14.
We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli–Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing. 相似文献
15.
Peter McMullen 《Geometriae Dedicata》1994,49(2):183-202
Pegged tilings localize the defining property of or Laguerre tilings, and, like them, admit a natural duality (corresponding to the Delaunay tilings of tilings). It can thus be shown that the projection method, which is generally used to construct quasi-periodic tilings related to tilings of higher dimensional lattices, applies to this wider class of tilings. Of further importance is that pegged tilings are just those which can be lifted to the graphs of convex functions with a certain strong locally polyhedrality property. The context of convex functions also provides a direct way of viewing the projection method, and leads to alternative pictures of special cases such as various grid methods. 相似文献
16.
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
17.
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
18.
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
19.
We consider one-dimensional quasiperiodic Fibonacci tilings. Namely, we study sets of vertices of these tilings that represent
one-dimensional quasilattices defined on the base of a parameterization by rotations of a circle, and the distribution of
points of quasilattices with respect to a variable module. We show that the distribution with respect to some modules is not
uniform. We describe the distribution function and its integral representation, and estimate the remainder in the problem
of the distribution of points of a quasilattice for corresponding modules. 相似文献
20.
This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive
ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems
on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings.
Received July 19, 2000, and in revised form January 30, 2001. Online publication July 25, 2001. 相似文献