Distances on Lozenge Tilings

Geometriae Dedicata, 1987

The paper discusses homeomorphic types of (periodic) filings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set ~ on which three involutions ao,al and tr2 act from the right such that troa ~ = a2ao and there are two maps mol ,m12 :~ ~ ~ satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Griinbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique,, are stated.

In this paper, we consider domino tilings of regions of the form D×[0,n], where D is a simply connected planar region and n∈N. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by ±1. We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.

2007

It is known that any two domino tilings of a polygon are flip-accessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flip-accessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, we provide three

Journal of Algebra, 2014

We show that for any prime number n = 2r +1 5 there exist r planar tilings with self-similar vertex set and the symmetry of a regular n-gon (D n-symmetry). The tiles are the rhombi with angle πk/n for k = 1,. .. , r.

Discrete & Computational Geometry, 1998

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite.

2008

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.

1998

We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.

For a given finite set of tiles and a strip of fixed height we describe how to obtain the associated (finite directed) graph that encodes the structure of the set of all possible tilings. We then consider possible constraints on such graphs and introduce and give some preliminary results on the reverse question: Given a graph, does there exist a set of tiles and a fixed height such that the corresponding set of tilings would generate the given graph? 1. Introduction The enumeration and classification of tilings has a long history. In 1953 Solomon Golomb introduced polyominoes, connected figures formed by squares placed so that each square shares a side with at least one other square (see [1], [2], and [3]). In 1965 Golomb's book on the subject, Polyominoes: Puzzles, Patterns, Problems and Packings [4], was published and contained many problems associated with polyominoes. One type of problem that has generated much research is the question of tiling rectangles with polyominoes. ...

Discrete and Computational Geometry, 2019

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.