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A tiling of $\mathbb {F}_{2}^{n}$ (considered as a graph) gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on $\mathbb {F}_{2}^{n}$ that are determined by a weight which respects the support of vectors (TS-metrics) and in this case, a perfect code is called TS-perfect. We consider all the tilings of $\mathbb {F}_{2}^{n}$ by tiles with up to 8 elements (which are classified in the literature) and determine which of them give rise to a TS-perfect code. In the sequence, for those tilings that give rise to a perfect code for some TS-metric, we classify the TS-metrics (up to equivalence) that turn it into a perfect code. Also, we give a general construction to obtain TS-perfect codes out of given TS-perfect codes, by proving that concatenation of two TS-perfect codes is by itself a TS-perfect code. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition on a d-dimensional affine subspace of R n to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of coincidence and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns. [ABSTRACT FROM AUTHOR]

In the famous paper (Brooks et al., 1940) an equivalence was established between planar electric networks and tilings of cylinders. However, the proof given there is rather difficult and requires knowledge of a larger theory. We give a simple new proof of this equivalence. Furthermore, there is a well known relationship between electric networks and random walks. We show how the tiling associated with the network corresponding to a random walk can shed light on various aspects of the walk, such as hitting times. [ABSTRACT FROM AUTHOR]

In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps $T$ on $\mathbb{R}^{d}$. We assume that (i)  $T$ consists of finitely many affine maps defined on a Borel measurable partition of  $\mathbb{R}^{d}$ , (ii) there is a lattice $\mathscr{L}\subset \mathbb{R}^{d}$ that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of  $\mathscr{L}$. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of $T$ are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact $T$ -invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for $T$ in the case of bounded single tiles. [ABSTRACT FROM AUTHOR]

We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group. Piantadosi [19] gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.We consider two other conditions: the first, also given by Piantadosi [19], is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al. [9], is a necessary condition to decide if a set of Wang tiles gives a tiling of Z2.We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel [14]. [ABSTRACT FROM AUTHOR]

This paper deals with the geometric analysis of the Roman mosaics found in the domūs of the archaeological areas of Volubilis, in Morocco, and Bulla Regia, in Tunisia. It analyzes the expressions of mosaic art which in the Roman Empire, between the second and fourth centuries A.D., attained a particular geometric decorative refinement. The tessellation of the mosaics of Volubilis and Bulla Regia is just an example of a cultural universe that for centuries has been a testimony to the lifestyles and myths of the Roman Empire and that today, if appropriately exploited, could contribute to enhance the knowledge and the identity of the Mediterranean heritage. The research proposes a reading of the mosaic apparatus through the study of the constitutive geometric elements. This leads to an analysis useful for the classification of figures and symmetry patterns, and understanding of their meanings. The methodology adopted is that of disassembly and reassembly of areas of mosaic tiling, creating a dialogue between the two-dimensional geometric elements of the flooring and the architectural context in which they are inserted. [ABSTRACT FROM AUTHOR]

We study the existence of Gabor orthonormal bases with window the characteristic function of the set Ω = [ 0 , α ] ∪ [ β + α , β + 1 ] of measure 1, with α , β > 0. By the symmetries of the problem, we can restrict our attention to the case α ≤ 1 / 2. We prove that either if α < 1 / 2 or (α = 1 / 2 and β ≥ 1 / 2) there exist such Gabor orthonormal bases, with window the characteristic function of the set Ω, if and only if Ω tiles the line. Furthermore, in both cases, we completely describe the structure of the set of time–frequency shifts associated to these bases. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of lattice perfect codes in two families of two-dimensional lattices built from the Minkowski embedding of the ring of integers of a quadratic field K = Q [ l ] , where l is a square-free integer. We analyzed the forms and quantities of perfect codes for two of these families lattices, one Orthogonal and the other Non-Orthogonal and we verified that we always managed to find a perfect code in these two families of lattices. The greater the value of l the more perfect codes we get. [ABSTRACT FROM AUTHOR]

We define a Wang tile set U of cardinality 19 and show that the set Ω U of all valid Wang tilings Z 2 → U is self-similar, aperiodic and is a minimal subshift of U Z 2 . Thus U is the second smallest self-similar aperiodic Wang tile set known after Ammann's set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable 2-dimensional morphism ω : Ω U → Ω U that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles. [ABSTRACT FROM AUTHOR]

We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the halting problem for deterministic cellular automata on the pentagrid. We prove that the problem is decidable when the cellular automaton starts its computation from a finite configuration and when it has two states, one of them being a quiescent state. [ABSTRACT FROM AUTHOR]

This paper defines a tiling problem related to the automorphism group of F 2 . Our main result is that the corresponding tilings admit a complete, concrete classification. [ABSTRACT FROM AUTHOR]

We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses geometric representations of stepped surfaces and related polygonal tilings, as well as self-replicating and periodic tilings made of Rauzy fractals. We apply our theory to one-parameter family of substitutions. For this family, we analyze and interpret in a new combinatorial way the codings of a domain exchange defined on the associated fractal domains. We deduce that the symbolic dynamical systems associated with this family of substitutions behave dynamically as first returns of toral translations. [ABSTRACT FROM AUTHOR]

In this paper, we show that non-symmetric convex polytopes cannot serve as a window function to produce a Gabor orthonormal basis by any time-frequency sets. [ABSTRACT FROM AUTHOR]

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using the theory of Diophantine Approximation, hence overcoming long-established difficulties, such as generalizations to higher-dimensional analogues. The universality of the method is demonstrated through its applications to different tiling problems. These include tiling rectangles with other rectangles, with their respective higher-dimensional counterparts, as well as tiling equilateral triangles, parallelograms, and trapezoids with equilateral triangles. [ABSTRACT FROM AUTHOR]

All possible combinatorial embeddings into primitive cubic networks of arbitrary tilings of 3D space by pairwise congruent and parallel regular hexagonal prisms are discussed and classified. [ABSTRACT FROM AUTHOR]

We study a problem about the number of lattice plane tilings by the given area centrosymmetrical polyominoes. A polyomino is a connected plane geomatric figure formed by joiining a finite number of unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a plane by polyominoes. In particular, the tiling is a lattice tiling if each tile can be mapped to any other tile by a translation which maps the whole tiling to itself. Earlier we proved that, for the number T(n) of a lattice plane tilings by polyominoes of an area n, holds the inequalities 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2, 7)n+1 . In the present work we prove a similar estimate for the number of lattice tilings with an additional central symmetry. Let Tc(n) be a number of lattice plane tilings by a given area centrosymmetrical polyominoes such that its translation lattice is a sublattice of Z 2 . It is proved that C1( √ 2)n ≤ Tc(n) ≤ C2n 2 ( √ 2.68)n . In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyominoes, and on the theory of self-avoiding walks on a square lattice.

We introduce a new four-parameter sequence that simultaneously generalizes some well-known integer sequences, including Fibonacci, Padovan, Jacobsthal, Pell, and Lucas numbers. Combinatorial interpretations are discussed and many identities for this general sequence are derived. As a consequence, a number of identities for Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Narayana numbers as well as some of their generalizations are obtained. We also present the Cassini formula for the new sequence. [ABSTRACT FROM AUTHOR]

The Hurwitz problem asks which ramification data are realizable , i.e., appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data. [ABSTRACT FROM AUTHOR]

We introduce the notion of degree of imperfection of a code in Zn with the ℓp metric, to extend the so-called quasi-perfect codes. Through the establishment of bounds and computational approach, we determine all radii for which there are linear quasi-perfect codes for p=2 and n=2,3. Numerical results concerning the codes with small degree of imperfection are also presented. [ABSTRACT FROM AUTHOR]

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set { x 1 , … , x n } of integers where x 1 < ⋯ < x n , let the gap sequence of this set be the unordered multiset { d 1 , … , d n − 1 } = { x i + 1 − x i : i ∈ { 1 , … , n − 1 } } . This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers p and q , there is a positive integer r 0 such that for all r ≥ r 0 , the set of integers can be partitioned into 4-sets with gap sequence p , q , r . [ABSTRACT FROM AUTHOR]

This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them. [ABSTRACT FROM AUTHOR]

We consider an orbit category of the bounded derived category of a path algebra of type A n which can be viewed as a − ( m + 1 ) -cluster category, for m ⩾ 1 . In particular, we give a characterisation of those maximal m -rigid objects whose endomorphism algebras are connected, and then use it to explicitly study these algebras. Specifically, we give a full description of them in terms of quivers and relations, and relate them with (higher) cluster-tilted algebras of type A . As a by-product, we introduce a larger class of algebras, called tiling algebras . [ABSTRACT FROM AUTHOR]

This paper studies stationary tessellations and tilings of the plane in which all cells are convex polygons. The focus is on the class of tessellations which are not side-to-side. The character of these tessellations is explored, with special attention paid to the relationship between edges of the tessellation and sides of the polygonal cells and to the combinatorial topology between the ‘adjacent’ geometric elements of the tessellation. Three new parameters, e0,e1 and e2 summing to unity, are introduced. These capture the essence of non side-to-side tessellations and play a role in understanding the adjacency of sides and cells. Examples illustrate the theory.

Beauty judgments for regular polygon tessellations were examined in two experiments. In experiment 1 we tested the three regular and eight semi-regular tilings characterized by a single vertex. In experiment 2 we tested the 20 demi-regular tilings containing two vertices. Observers viewed the tessellations at different random orientations inside a circular aperture and rated them using a numeric 1−7 scale. The data from the first experiment show a peak in preference for tiles with two types of polygons and for five polygons around a vertex. Triangles were liked more than other geometric shapes. The results from the second experiment demonstrate a preference for tessellations with a greater number of different kinds of polygons in the overall pattern and for tiles with the greatest difference in the number of polygons between the two vertices. Ratings were higher for tiles with circular arrangements of elements and lower for those with linear arrangements. Symmetry group p6m was liked the most and groups cmm and pmm were liked the least. Taken as a whole the results suggest a preference for complexity and variety in terms of both vertex qualities and symmetric transformations. Observers were sensitive to both the underlying mathematical properties of the patterns as well as their emergent organization.