A short proof of two shuffling theorems for tilings and a weighted generalization

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.     

A simple explanation for the “shuffling phenomenon” for lozenge tilings of dented hexagons

In a recent preprint, Lai and Rohatgi proved a “shuffling theorem” for lozenge tilings of a hexagon with “dents” (i.e., missing triangles). Here, we shall point out that this follows immediately from the enumeration of Gelfand–Tsetlin patterns with given bottom row. This observation is also contained in a recent preprint of Byun.     

Orbits of edge-to-edge tilings by equilateral triangles,squares and regular hexagons

An edge-to-edge tiling of the Euclidian plane by equilateral triangles, squares and regular hexagons is said to be of type (t,s,h) if there are exactly t orbits of triangles, s orbits of squares and h orbits of hexagons under the symmetry group of the tiling. We prove that there exist tilings of type (t,s,h) for every t 92, s 2, h 43. We completely determine the values of t and h for which tilings of type (t,1,h) exist.     

Incongruent equipartitions of the plane

R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the problem, or for the analogue of this problem for quadrangles, pentagons, or hexagons. Several answers were given by the first author in a previous paper. Here we solve three further cases. In particular, our main result shows that there are vertex-to-vertex tilings by pairwise incongruent triangles of unit area and bounded perimeter.     

The Cubical Matching Complex

Given a bipartite planar graph embedded in the plane, we define its cubical matching complex. By combining results of Kalai and Propp, we show that the cubical matching complex is collapsible. As a corollary, we obtain that a simply connected region R in the plane that can be tiled with lozenges and hexagons satisfies ${f_0 - f_1 + f_2 - \cdots = 1}$ , where f _i is the number of tilings with i hexagons. The same relation holds for a region tiled with dominoes and 2 × 2 squares. Furthermore, we show for a region that can be tiled with dominoes, that each link of the associated cubical complex ${\mathcal{C}(R)}$ is either collapsible or homotopy equivalent to a sphere.     

Self-generating hexagonal cell division patterns

Consider the plane covered by regular hexagons. We investigate division patterns in which each hexagon is divided into two new regions, each new region has six neighbouring regions and each vertex in the new structure belongs to three new regions. These patterns are of interest for cell division processes in biology and are related to a certain class of hexagonal tilings of the plane.Research supported in part by the University of Utrecht and in part by the OTKA Research Fund of the Hungarian Academy of Sciences.     

Rhombus Tilings: Decomposition and Space Structure

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.     

Duality properties between spectra and tilings

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.     

Convex pentagons that admit <Emphasis Type="Italic">i</Emphasis>-block transitive tilings

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.     

Some 2-isohedral tilings of the plane

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.     

σ-Morphic sets of prototiles

Using a general construction method, sets of prototiles (with more than one element) are obtained which admit a countable infinity of distinct tilings. In contrast to the tilings described in a previous paper, in this case almost all the tilings are periodic. In particular, sets of two, three, four, and five prototiles are described.     

Duality,sections and projections of certain Euclidean tilings

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.     

Ammann bars and quasicrystals

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.     

A note on the structure of spaces of domino tilings

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.     

One-dimensional quasiperiodic tilings admitting progressions enclosure

In this paper we consider one-dimensional quasiperiodic tilings based on the use of irrational rotations of a circle. We completely describe a wide class of progressions included in the mentioned tilings.     

Generating Function of the Tilings of an Aztec Rectangle with Holes

We consider a generating function of the domino tilings of an Aztec rectangle with several unit squares removed from the boundary. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.     

Monotone labelings in polygonal tilings

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.     

Parallelogram Tilings, Worms, and Finite Orientations

This paper studies properties of tilings of the plane by parallelograms. In particular, it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.