Number Of Tiles Research Articles

Tiling Parity Results and the Holey Square Solution

We prove combinatorially that the parity of the number of domino tilings of a region is equal to the parity of the number of domino tilings of a particular subregion. Using this result we can resolve the holey square conjecture. We furthermore give combinatorial proofs of several other tiling parity results, including that the number of domino tilings of a particular family of rectangles is always odd.

Classification of Voronoi and Delone tiles of quasicrystals: III. Decagonal acceptance window of any size

This paper is the last of a series of three articles presenting a classification of Vornoi and Delone tilings determined by point sets ?(?) ('quasicrystals'), built by the standard projection of the root lattice of type A4 to a two-dimensional plane spanned by the roots of the Coxeter group H2 (dihedral group of order 10). The acceptance window ? for ?(?) in the present paper is a regular decagon of any radius 0 < r < ?. There are 14 distinct VT sets of Voronoi tiles and 6 sets DT of Delone tiles, up to a uniform scaling by the factor and . The number of Voronoi tiles in different quasicrystal tilings varies between 3 and 12. Similarly, the number of Delone tiles is varying between 4 and 6. There are 7 VT sets of the 'generic' type and 7 of the 'singular' type. The latter occur for seven precise values of the radius of the acceptance window. Quasicrystals with acceptance windows with radii in between these values have constant VT sets, only the relative densities and arrangement of the tiles in the tilings change. Similarly, we distinguish singular and generic sets DT of Delone tiles.

Determinant formulae for some tiling problems and application to fully packed loops

We present a number of determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration.

Innovative substrate tiles to spatially control larval settlement in coral culture

We designed settlement tiles for coral culture with the aims: (1) to attract and (2) to spatially control settlement and (3) to produce substrate units for further handling in mariculture. Ceramic tiles with vertical (i.e. pyramid tile; truncated pyramid shape, 22.0×22.0, ±1.0 mm, L×W) and horizontal surfaces (i.e. flat tile; 17.0×17.0, ±1.0 mm, L×W) were temporarily arranged chessboard-like in a polystyrene grid. Surfaces of both types were structured with grooves (width and depth 2.0±0.1 mm). Material analysis by ICP and photospectrometry showed negative influences on water chemistry when new tiles were directly used; high contents of heavy metals could be detected depending on the production process. Planulae of Acropora tenuis (16.3±3.1% settlement) preferred to settle in the grooves of pyramid tiles with a density of 0.19±0.03 to 0.44±0.06 settlers cm−2, whereas those of Favia fragum (60.7±13.3% settlement) showed high preferences for the grooves of flat tiles (0.48±0.22 to 1.05±0.44 settlers cm−2). Settlement occurred exclusively on surfaces with a biofilm (turf and crustose coralline algae) and was independent of available substrate surface area, whereas the number of tiles per set-up influenced the density of settlers per unit. Spatial control of settlement was achieved through tile design and arrangement. Tiles with settlers could be easily separated for further use in coral culture.

On the chromatic number of plane tilings

AbstractIt is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.

Tilings of rectangles with T-tetrominoes

We prove that any two tilings of a rectangular region by T-tetrominoes are connected by moves involving only 2 and 4 tiles. We also show that the number of such tilings is an evaluation of the Tutte polynomial. The results are extended to a more general class of regions.

Symmetry groups for beta-lattices

We present a construction of symmetry plane-groups for quasiperiodic point-sets named beta-lattices. The framework is issued from beta-integers counting systems. Beta-lattices are vector superpositions of beta-integers. When β >1 is a quadratic Pisot–Vijayaraghavan algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When β = (1+ 5 ) 2 , 1+ 2 and 2+ 3 , we show that these arithmetic and algebraic structures lead to freely generated symmetry plane-groups for beta-lattices. These plane-groups are based on repetitions of discrete adapted rotations and translations we shall refer to as “beta-rotations” and “beta-translations”. Hence beta-lattices, endowed with beta-rotations and beta-translations, can be viewed like lattices. The quasiperiodic function ρ S ( n ), defined on the set of beta-integers as counting the number of small tiles between the origin and the n th beta-integer, plays a central part in these new group structures. In particular, this function behaves asymptotically like a linear function. As an interesting consequence, beta-lattices and their symmetries behave asymptotically like lattices and lattice symmetries, respectively.

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel–Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.

Symmetry groups for beta-lattices

We present a construction of symmetry plane-groups for quasiperiodic point-sets named beta-lattices. The framework is issued from beta-integers counting systems. Beta-lattices are vector superpositions of beta-integers. When β>1 is a quadratic Pisot–Vijayaraghavan algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When β = (1+ 5 ) 2 , 1+ 2 and 2+ 3 , we show that these arithmetic and algebraic structures lead to freely generated symmetry plane-groups for beta-lattices. These plane-groups are based on repetitions of discrete adapted rotations and translations we shall refer to as “beta-rotations” and “beta-translations”. Hence beta-lattices, endowed with beta-rotations and beta-translations, can be viewed like lattices. The quasiperiodic function ρ S ( n), defined on the set of beta-integers as counting the number of small tiles between the origin and the nth beta-integer, plays a central part in these new group structures. In particular, this function behaves asymptotically like a linear function. As an interesting consequence, beta-lattices and their symmetries behave asymptotically like lattices and lattice symmetries, respectively.

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel–Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.

Tilings of rectangles with T-tetrominoes

We prove that any two tilings of a rectangular region by T-tetrominoes are connected by moves involving only 2 and 4 tiles. We also show that the number of such tilings is an evaluation of the Tutte polynomial. The results are extended to a more general class of regions.

Approximation algorithms for MAX–MIN tiling

The MAX–MIN tiling problem is as follows. We are given A [1,…,n][1,…,n] , a two-dimensional array where each entry A [i][j] stores a non-negative number. Define a tile of A to be a subarray A [ℓ,…,r][t,…,b] of A , the weight of a tile to be the sum of all array entries in it, and a tiling of A to be a collection of tiles of A such that each entry A [i][j] is contained in exactly one tile. Given a weight bound W the goal of our MAX–MIN tiling problem is to find a tiling of A such that: (1) each tile is of weight at least W (the MIN condition), and (2) the number of tiles is maximized (the MAX condition). The MAX–MIN tiling problem is known to be NP-hard; here, we present first non-trivial approximations algorithms for solving it.

Counting domains in { p, q} tessellations

For any given regular { p, q} tessellation in the hyperbolic plane, we compute the number of vertices and tiles to be found as we distance from a given point, enabling a complete characterization of the asymptotic behavior.

Implementing data cube construction using a cluster middleware: algorithms, implementation experience, and performance evaluation

With increases in the amount of data available for analysis in commercial settings, on line analytical processing (OLAP) and decision support have become important applications for high performance computing. Implementing such applications on clusters requires a lot of expertise and effort, particularly because of the sizes of input and output datasets. In this paper, we describe our experiences in developing one such application using a cluster middleware, called ADR. We focus on the problem of data cube construction, which commonly arises in multi-dimensional OLAP. We show how ADR, originally developed for scientific data intensive applications, can be used for carrying out an efficient and scalable data cube construction implementation. A particular issue with the use of ADR is tiling of output datasets. We present new algorithms that combine interprocessor communication and tiling within each processor. These algorithms preserve the important properties that are desirable from any parallel data cube construction algorithm. We have carried out a detailed evaluation of our implementation. The main results from our experiments are as follows: (1) high speedups are achieved on both dense and sparse datasets, even though we have used simple algorithms that sequentialize a part of the computation; (2) the execution time depends only upon the amount of computation, and does not increase in a super-linear fashion as the dataset size or the number of tiles increases; and (3) as the datasets become more sparse, sequential performance degrades, but the parallel speedups are still quite good. As part of our on-going work in this area, we are also looking at handling a larger number of dimensions and multi-dimensional partitionings. We describe our preliminary theoretical and experimental work in this direction.

Beryllium and carbon films in JET following D–T operation

After the D–T operation (DTE-1 campaign) at JET a large number of limiter and divertor tiles were dismounted from the torus for ex situ examination. The relative distributions of deuterium, tritium, beryllium and carbon are presented and discussed. Significant asymmetry observed in the distribution of erosion and deposition zones indicates preferential flow of the deuterium background plasma and impurities towards the inner divertor leg. The comparison of the beryllium content on the limiter tiles from the main chamber and the content of this element on the inner divertor tiles clearly proves the beryllium erosion from the main chamber wall and its transport to the divertor. However, no beryllium is detected in the shadowed regions of the divertor where the formation of thick and fuel-rich carbon films occurs. This is interpreted in terms of different mechanisms governing the erosion and transport of Be and C. The results allow a conclusion that the operation with a full beryllium wall would lead to a significantly decreased fuel inventory due to removal of the carbon source.

Perfect Matchings and Perfect Powers

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111–132s B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991s J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version.

Enumeration of Lozenge Tilings of Hexagons with Cut-Off Corners

Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases” removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.

The Kernel of the Adjacency Matrix of a Rectangular Mesh

Given an m × n rectangular mesh, its adjacency matrix A , having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K . We describe the kernel of A : it is a direct sum of two natural subspaces whose dimensions are equal to $\lceil c/2 \rceil$ and $\lfloor c/2 \rfloor$ , where c = gcd (m+1,n+1) - 1 . We show that there are bases to both vector spaces, with entries equal to 0,1 and -1 . When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles.

A Non-Automatic (!) Application of Gosper's Algorithm Evaluates a Determinant from Tiling Enumeration

We evaluate the determinant det 1≤i,j≤n x (() - ()), which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it requires the proof of a certain hypergeometric identity which we accomplish by using Gosper's algorithm in a nonautomatic fashion.

Use of tile overlays to reduce the radiation efficiency of radiating surfaces

An intriguing possibility for reducing the radiation efficiency of a structure is to overlay its radiating surfaces with a contiguous set of low mass tiles. Experimental results show that significant reductions in sound power are possible depending on number of tiles, their geometry, attachment methods, mass, stiffness, and the amount of shear damping used to connect their edges to one another. This paper presents results of a combined theoretical and experimental study performed on a clamped, flat plate driven at its center and with nearly infinite baffle boundary conditions. Tiles are overlayed on the surface of the plate and lightly attached with a polymer material. A scanning laser vibrometer is used to examine the vibration patterns of the plate and the corresponding tiles for several different tile configurations over a frequency range 20–1000 Hz. Results indicate that sound power reductions occur when the independent response of each tile combines to produce an overall volume velocity that is lower than that of the plate by itself over the frequency range of interest.