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A family of spherical caps of the 2-dimensional unit sphere S 2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S 2 . We solve this problem up to eight spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of unit balls in Euclidean 3-space is 8. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n > 1 great circles in S 2 . Next, we prove dual bounds for TS-coverings of S 2 by congruent spherical caps. Here a covering of S 2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S 2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension > 2 . [ABSTRACT FROM AUTHOR]

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a 3 b ). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings. [ABSTRACT FROM AUTHOR]

The kisrhombille tiling is the dual tessellation of one of the semi‐regular tessellations. It consists of right‐angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons. [ABSTRACT FROM AUTHOR]

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n ≥ 5 . [ABSTRACT FROM AUTHOR]

We study the problem of perfect tiling in the plane and explore the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper. [ABSTRACT FROM AUTHOR]

Quasiperiodic patterns and crystals—having long range order without translational symmetry—have fascinated researchers since their discovery. In this study, we report on new p‐terphenyl‐based T‐shaped facial polyphiles with two alkyl end chains and a glycerol‐based hydrogen‐bonded side group that self‐assemble into an aperiodic columnar liquid quasicrystal with 12‐fold symmetry and its periodic liquid‐crystalline approximants with complex superstructures. All represent honeycombs formed by the self‐assembly of the p‐terphenyls, dividing space into prismatic cells with polygonal cross‐sections. In the perspective of tiling patterns, the presence of unique trapezoidal tiles, consisting of three rigid sides formed by the p‐terphenyls and one shorter, incommensurate, and adjustable side by the alkyl end chains, plays a crucial role for these phases. A delicate temperature‐dependent balance between conformational, entropic and space‐filling effects determines the role of the alkyl chains, either as network nodes or trapezoid walls, thus resulting in the order‐disorder transitions associated with emergence of quasiperiodicity. In‐depth analysis suggests a change from a quasiperiodic tiling involving trapezoids to a modified one with a contribution of trapezoid pair fusion. This work paves the way for understanding quasiperiodicity emergence and develops fundamental concepts for its generation by chemical design of non‐spherical molecules, aggregates, and frameworks based on dynamic reticular chemistry. [ABSTRACT FROM AUTHOR]

In this paper we study uniformly random lozenge tilings of strip domains. Under the assumption that the limiting arctic boundary has at most one cusp, we prove a nearly optimal concentration estimate for the tiling height functions and arctic boundaries on such domains: with overwhelming probability the tiling height function is within n δ of its limit shape, and the tiling arctic boundary is within n 1 / 3 + δ to its limit shape, for arbitrarily small δ > 0 . This concentration result will be used in Aggarwal and Huang (Edge statistics for lozenge tilings of polygons, II: Airy line ensemble, 2021. arXiv:2108.12874) to prove that the edge statistics of simply-connected polygonal domains, subject to a technical assumption on their limit shape, converge to the Airy line ensemble. [ABSTRACT FROM AUTHOR]

The escalating demand for heat dissipation has prompted considerable interest in the manifold micro-pin–fin heat sink (MMPFHS) as a promising solution for high heat flux cooling applications. The two-dimensional topology of this multi-layered heat sink plays a pivotal role in organizing the three-dimensional spatial flow, consequently influencing both hydraulic and thermal performance. This study proposes and comparatively analyzes the only three rhombus-based tiling topologies applicable to MMPFHS, specifically the rhombus topology (RT), triangle hexies topology (THT), and firecracker topology (FT), all of which adhere to the principles of gapless, non-overlapping, and spatially expandable tiling topology. The relations of topological geometry and the hydraulic and thermal performances are explored and compared. And figure of merit (FOM) is also introduced to assess their overall performance. The results indicate that the overall performance of THT is predominant. When DPF = 100 μm, a = 200 μm, Din = 50, Re = 64.87, the temperature non-uniformity on the heating surface of THT is only 0.022 K, the thermal resistance is 9.4 × 10–6 km2 W−1, and the FOM of THT is about 1.14 times of that of RT. RT and FT have a similar performance, and the maximum deviation of FOM for both of them within the conditions studied in this paper is only 0.04. The paper provides a reference for selecting topologies in MMPFHS design. [ABSTRACT FROM AUTHOR]

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation‐reversing three‐interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We observe that, for a particular choice of triangle tiling, certain trajectories appear to approach the Rauzy fractal, under rescaling. [ABSTRACT FROM AUTHOR]

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ΓX of any manifold X. The name comes from the fact that various elements of the geometry of ΓX are constructed via lifting of the corresponding elements of the geometry of X. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various "infinite dimensional spaces" associated to X. In order to define a lifted geometry for a "space", one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on X, mappings into X, embedded submanifolds of X, and tilings on X, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a random measure is discussed. It is shown that Stokes' Theorem appears as "differentiability" of "boundary operator" in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on X form the algebra of smooth functions in a specific lifted geometry for the path-space of X. [ABSTRACT FROM AUTHOR]

First, we consider order- n ribbon tilings of an M -by- N rectangle R M , N where M and N are much larger than n. We prove the existence of the growth rate γ n of the number of tilings and show that γ n ≤ (n − 1) ln 2. Then, we study a rectangle R M , N with fixed width M = n , called a strip. We derive lower and upper bounds on the growth rate μ n for strips as ln n − 1 + o (1) ≤ μ n ≤ ln n. Besides, we construct a recursive system which enables us to enumerate the order- n ribbon tilings of a strip for all n ≤ 8 and calculate the corresponding generating functions. [ABSTRACT FROM AUTHOR]

We consider measurable functions f$f$ on R$\mathbb {R}$ that tile simultaneously by two arithmetic progressions αZ$\alpha \mathbb {Z}$ and βZ$\beta \mathbb {Z}$ at respective tiling levels p$p$ and q$q$. We are interested in two main questions: what are the possible values of the tiling levels p,q$p,q$, and what is the least possible measure of the support of f$f$? We obtain sharp results which show that the answers depend on arithmetic properties of α,β$\alpha , \beta$ and p,q$p,q$, and in particular, on whether the numbers α,β$\alpha , \beta$ are rationally independent or not. [ABSTRACT FROM AUTHOR]

Digital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave‐front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose an integer linear programming model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how it can be used to define an iterative algorithm that, given a period n, finds all the rhythms which tile with a given rhythm A and also to efficiently check the necessity of the Coven-Meyerowitz condition (T2). To conclude, we run several experiments to validate the time efficiency of the model. [ABSTRACT FROM AUTHOR]

The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. Pólya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk. [ABSTRACT FROM AUTHOR]

A self-affine tile is a compact set G ⊂ R d that admits a partition (tiling) by parallel shifts of the set M - 1 G , where M is an expanding matrix. We find all self-affine tiles which are polyhedral sets, i.e., unions of finitely many convex polyhedra. It is shown that there exists an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to integral self-affine tiles with standard digit sets, when the matrix M and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed. [ABSTRACT FROM AUTHOR]

We study self-affine tiles generated by iterated function systems consisting of affine mappings whose linear parts are defined by different matrices. We obtain an interior theorem for these tiles. We prove a tiling theorem by showing that for such a self-affine tile, there always exists a tiling set. We also obtain a more complete interior theorem for reptiles, which are tiles obtained when the matrices in the iterated function system are similarities. Our results extend some of the classical ones by Lagarias and Wang (Adv. Math. 121(1), 21–49 (1996)), where the IFS maps are defined by a single matrix. [ABSTRACT FROM AUTHOR]

We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. [ABSTRACT FROM AUTHOR]

A d-dimensional configuration c : Z d ⟶ A is a coloring of the d-dimensional infinite grid by elements of a finite alphabet A ⊆ Z . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a (d - 1) -dimensional linear subspace S ⊆ R d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all (d - 1) -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of Z d by translations of a single tile. [ABSTRACT FROM AUTHOR]

In the famous works "Tilings, substitution systems and dynamical systems generated by them" by S. Mozes and "Matching rules and substitution tilings" by Ch. Goodman-Strauss, important properties of tiling on the plane connecting the language of substitution systems and local rules are investigated. In particular, it is proved that with almost any substitution system of tiles, it is possible to link a system of decorations and local rules to the boundaries of tiles in such a way that any tiling of the plane allowed by the rules will belong to the family generated by this substitution system. Geometric constructions based on tiling and geometric complexes of glued squares open up new opportunities for the study and construction of algebraic objects. An important property of tiling that allows them to be associated with algebraic objects is determinicity. For square tiles whose sides are painted in a finite number of colors, this property means that the colors of the two adjacent sides of each tile uniquely determine the colors of the other pair of sides. J. Kari and P. Papasoglu constructed a deterministic aperiodic set. Thus, the analogue of the theorems of Goodman-Strauss and Mozes with an additional determinicity condition is an important statement in the theory of aperiodic mosaics. We prove an analog of this statement for a specific fixed substitution system. But its arbitrariness allows us to hope for the possibility of improving the construction for the general case. This system is important when constructing an infinite finitely presented nilsemigroup, which solves the problem of L. N. Shevrin and M. V. Sapir. The work is also devoted to the construction of a family of geometric complexes and the introduction of coding of vertices and edges on them. For the resulting complex, both properties are fulfilled simultaneously: finiteness of the set of defining local rules and determinicity on paths of length 2 on minimal squares. [ABSTRACT FROM AUTHOR]

We consider the five-vertex model on a regular square lattice of the size L × M with boundary conditions fixed in such a way that configurations of the model are in one-to-one correspondence with the lozenge tilings of the hexagon with a dent. We obtain two determinant representations for the partition function. In the free-fermionic limit, this result implies some summation formulae for Schur functions. [ABSTRACT FROM AUTHOR]

In this paper we study extremal behaviors of the mean to max ratio of the p-torsion function with respect to the geometry of the domain. For p larger than the dimension of the space N, we prove that the upper bound is uniformly below 1, contrary to the case p ∈ (1 , N ] . For p = + ∞ , in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size. [ABSTRACT FROM AUTHOR]

In 1966 Gallai asked the following question: Do all longest paths (cycles) of a connected graph contain a common vertex? After a positive answer to Gallai's question, another question has been raised, Is there any family of graphs without Gallai's property? Menke found one such family, the square lattices. Embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and often used in quantum mechanics and material sciences. In this paper, we prove the existence of graphs with the empty intersection of their longest cycles as subgraphs of Archimedean lattices. [ABSTRACT FROM AUTHOR]

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision. [ABSTRACT FROM AUTHOR]

Given two graphs H and G, an H-tiling of G is a collection of vertex-disjoint copies of H in G and an H-factor is an H-tiling that covers all vertices of G. K\"uhn and Osthus managed to characterize, up to an additive constant, the minimum degree threshold which forces an H-factor in a host graph G. In this paper we study a similar tiling problem in a system that is locally bounded. An incompatibility system (G,\scrF) consists of a graph G and a family F = {Fv}v∊V(G) over G with Fv... We say that two edges e, et ∊ E(G) are incompatible if\{e,et} ∊ Fv for some v ∊ V (G), and otherwise compatible. A subgraph H of G is compatible if every pair of edges in H are compatible. An incompatibility system (G,F) is Δ-bounded if for any vertex v and any edge e incident with v, there are at most Δ members of Fv containing e. This notion was partly motivated by a concept of transition system introduced by Kotzig [Matematický časopis, 18 (1968), pp. 76--80], and first formulated by Krivelevich, Lee, and Sudakov [Combinatorica, 37 (2017), pp. 697--732], to study the robustness of Hamiltonicity of Dirac graphs. We prove that for any α >0 and any graph H with h vertices, there exists a constant mu >0 such that for any sufficiently large n with n ∊ hN, if G is an n-vertex graph with Δ (G)... and (G,F) is a\mu n-bounded incompatibility system, then there exists a compatible H-factor in G, where the value\chi\ast (H) is either the chromatic number χ (H) or the critical chromatic number χ cr(H) and we provide a dichotomy as in the K\"uhn--Osthus result. Moreover, we give examples H for which there exists an\mu n-bounded incompatibility system (G,F) with n ∊ hN and Δ (G)...n such that G contains no compatible H-factor. Unlike in the previous work of K\"uhn and Osthus on embedding H-factors, our proof uses the lattice-based absorption method. [ABSTRACT FROM AUTHOR]

Considerable energy has been devoted to understanding domino tilings: for example, Elkies, Kuperberg, Larsen and Propp proved the Aztec diamond theorem, which states that the number of domino tilings for the Aztec diamond of order n is equal to 2 n (n + 1) / 2 , and the authors recently counted the number of domino tilings for augmented Aztec rectangles and their chains by using Delannoy paths. In this paper, we count domino tilings for two new shapes of regions, bounded augmented Aztec rectangles and Aztec octagons by constructing a bijection between domino tilings for these regions and the associated generalized Motzkin paths. [ABSTRACT FROM AUTHOR]

We investigate Conway's Game of Life played on the Robinson triangle Penrose tiling. In this paper, we classify all four-cell still lifes. [ABSTRACT FROM AUTHOR]

There are various tessellations of the plane, including three regular and eight semi‐regular tilings. The square grid is self‐dual, and the hexagonal and triangular tilings are dual to each other. The semi‐regular tessellations are based on more than one type of regular tiles, while their dual tilings are based on a sole but not a regular tile. In various applications, including Geographical Information Systems, it is worth considering non‐regular grids instead of the most used square grid. In this article, we are interested in the dual of the semi‐regular truncated quadrille tiling, T(8,8,4), which is also known as the Khalimsky grid due to its connectedness structure. In our grid, which is called the tetrakis square or kisquadrille tiling, while it is denoted by D(8,8,4), we consider the right‐angled triangle regions of the usual two‐dimensional Khalimsky graph as tiles/pixels. We give an easy‐to‐use coordinate frame addressing the triangles of all the four different orientations. Neighbor relations are described mathematically based on this frame. Based on the shortest path algorithm, a closed formula is proven to compute the digital, that is, path‐based distance on this grid. Some properties of the distance function have also been studied. Hierarchical coarsening is a frequently used technique both in Geometric and Geographical Information Systems to rescale some parts of the map. The tetrakis square grid is apt for hierarchical coarsening, and thus, it can easily be used in image compression and multigrid and other related methods. [ABSTRACT FROM AUTHOR]

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision. [ABSTRACT FROM AUTHOR]

Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a 1/2 approximation guarantee for arbitrary tile sets, and a 2/3 guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provide very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic. [ABSTRACT FROM AUTHOR]

Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience. [ABSTRACT FROM AUTHOR]

A brief introductory review is provided of the theory of tilings of 3‐periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal‐transitivity tilings of nets are described. Essential rings are used for finding the minimal‐transitivity tiling for a given net. Tiling theory is used to find all edge‐ and face‐transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal‐transitivity tilings. This work identifies the 3‐periodic surfaces defined by the nets of the tiling and its dual and indicates how 3‐periodic nets arise from tilings of those surfaces. [ABSTRACT FROM AUTHOR]

We regard a smooth, d = 2 -dimensional manifold M and its normal tiling M, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by v ¯ ⋆ and we prove that if M is periodic then v ¯ ⋆ ≥ 2 . We show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with v ¯ ⋆ = 0 . [ABSTRACT FROM AUTHOR]

We introduce a graph connectedness induced by a given set of paths of the same length. We focus on the 2-adjacency graph on the digital line ℤ with a certain set of paths of length n for every positive integer n. The connectedness in the strong product of two and three copies of the graph is used to define digital Jordan curves and digital Jordan surfaces, respectively. Such definitions build on an edge-to-edge tiling with triangles in the digital plane and a face-to-face tiling by cubes, prisms and pyramids in the (3D) digital space, respectively. [ABSTRACT FROM AUTHOR]

In this paper we study colorings (or tilings) of the two-dimensional grid ℤ 2 . A coloring is said to be valid with respect to a set P of n × m rectangular patterns if all n × m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist m , n ∈ ℕ and a set P of n × m rectangular patterns such that c is valid with respect to P and |P|≤ nm. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, all valid colorings with respect to P such that |P|≤ mn must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat's conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that the nm bound is multiplicatively optimal for the decidability of the domino problem, as for all ε > 0 it is undecidable to determine if there exists a valid coloring for a given m , n ∈ ℕ and set of rectangular patterns P of size n × m such that |P|≤ (1 + ε)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 12). [ABSTRACT FROM AUTHOR]

By rationally adjusting the weaving modes of point‐star tiles, the curvature inherent in the tiles can be changed, and various DNA nanostructures can be assembled, such as planar wireframe meshes, perforated wireframe tubes, and curved wireframe polyhedra. Based on the weaving and tiling architectures for traditional point‐star tiles with the core arm length at two DNA half‐turns, we improved the weaving modes of our newly reported four‐point‐star tiles with the core arm length at three half‐turns to adjust their curvature and rigidity for assembling 2D arrays of DNA grids and tubes. Following our previous terms and methods to analyze the structural details of E‐tiling tubes, we used the chiral indices (n,m) to describe the most abundant tube of typical assemblies; especially, we applied both one‐locus and/or dual‐locus biotin/streptavidin (SA) labelling strategies to define the configurations of two specific tubes, along with the absolute conformations of their component tiles. Such structural details of the DNA tubes composed of tiles with addressable concave and convex faces and packing directions should help us understand their physio‐chemical and biological properties, and therefore promote their applications in drug delivery, biocatalysis, biomedicine, etc. [ABSTRACT FROM AUTHOR]

We study the asymptotic limit of random pure dimer coverings on rail yard graphs when the mesh sizes of the graphs go to 0. Each pure dimer covering corresponds to a sequence of interlacing partitions starting with an empty partition and ending in an empty partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the limit shape (law of large numbers) of the rescaled height functions and the convergence of the unrescaled height fluctuations to a diffeomorphic image of the Gaussian free field (Central Limit Theorem), answering a question in [7]. Applications include the limit shape and height fluctuations for pure steep tilings [9] and pyramid partitions [20; 36; 39; 38]. The technique to obtain these results is to analyze a class of Macdonald processes which involve dual partitions as well. [ABSTRACT FROM AUTHOR]

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to efficiently analyze the average behavior of certain common nonlinear dynamical systems, such as the angle-doubling map, x ↦ 2 x modulo 1. In particular, one can use this information to analyze how the behavior of individual orbits deviates from the global average (called fluctuations). More generally, the formula is valid in R m , where expanding maps give rise to so-called number systems. To illustrate the usefulness in this setting, we compute the fluctuations of a certain map on the plane. [ABSTRACT FROM AUTHOR]

Recently there has been significant progress in classifying integer friezes and \mathrm {SL}_2-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame \mathrm {SL}_2-tilings, positive integer \mathrm {SL}_2-tilings, and tame integer friezes – both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes which generalises the classical construction of Conway and Coxeter for positive integer friezes. [ABSTRACT FROM AUTHOR]

We use the periodic Schur process, introduced in (Borodin in Duke Math J 140(3):391–468 2007), to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the q vol measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes. [ABSTRACT FROM AUTHOR]

Motivated by a conjecture posed by Fulmek and Krattenthaler, we provide product formulas for the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion. As a direct corollary, we obtain a product formula for the number of boxed plane partitions with a certain restriction. We also investigate the asymptotic behavior of the ratio between the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion and that of a semiregular hexagon without an intrusion. [ABSTRACT FROM AUTHOR]

• The cubic grid Z 3 plays important role in the (digital) world. • Meshes obtained by cutting Z 3 by oblique planes can display 2d images. • The dual semi-regular grid, the rhombille tiling is an oblique cubic mesh. • Shortest path algorithm and digital distance are computed on the rhombille tiling. • The rhombille tiling can be used as a digital canvas and for RGB pixel geometry. In this paper, we investigate digital geometry on the rhombille tiling, D(6,3,6,3), that is the dual of the semi-regular tiling called hexadeltille T(6,3,6,3) tiling and also known as trihexagonal tiling. In fact, this tiling can be seen as an oblique mesh of the cubic grid giving practical importance to this specific grid both in image processing and graphics. The properties of the coordinate systems used to address the tiles are playing crucial roles in the simplicity of various algorithms and mathematical formulae of digital geometry that allow to work on the grid in image processing, image analysis and computer graphics, thus we present a symmetric coordinate system. This coordinate system has a strong relation to topological/combinatorial coordinate system of the cubic grid. It is an interesting fact that greedy shortest path algorithm may not be used on this grid, despite to this, we present algorithm to provide a minimal-length path between each pair of tiles, where paths are defined as sequences of neighbor tiles (those are considered to be neighbors which share a side). We also prove closed formula for computing the digital, i.e., path-based distance, the length (the number of steps) of a/the shortest path(s). Some example pictures on this grid are also presented, as well as its possible application as pixel geometry for color images and videos on the hexagonal grid. To create your abstract, type over the instructions in the template box below. Fonts or abstract dimensions should not be changed or altered. [Display omitted] [ABSTRACT FROM AUTHOR]

It is well known that the functions f 2 L¹.Rd whose translates along a lattice ƒ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set ƒ R (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions f; g 2 L¹. whose Fourier transforms have the same set of zeros, but such that f C ƒ is a tiling while g C ƒ is not. [ABSTRACT FROM AUTHOR]

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted \mathcal {T}_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions \mathcal {T}_n, we construct a family C_{m,n}^{a,b,c,d} of cruciform regions generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of \mathcal {T}_n is a divisor of the number of tilings of the cruciform region C_{2n-1,2n-1}^{n-1,n,n,n-2}, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture. [ABSTRACT FROM AUTHOR]

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain 'regular' covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets. [ABSTRACT FROM AUTHOR]

Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut‐and‐project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600–610] is that unimodularity is no longer assumed in this paper. [ABSTRACT FROM AUTHOR]