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The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat--the equilateral member of the continuum to which it belongs--is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only homochiral non-periodic tilings based on a hierarchical substitution system.Mathematics Subject Classifications: 05B45, 52C20, 05B50Keywords: Tilings, aperiodic order, polyforms

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical--and hence aperiodic--tilings.Mathematics Subject Classifications: 05B45, 52C20, 05B50Keywords: Tilings, aperiodic order, polyforms

In spring 2022, the Galileo Galilei Institute for Theoretical Physics hosted a sevenweek Workshop on 'Randomness, Integrability and Universality'. The Workshop addressed a series of questions on exactly solvable models of statistical mechanics, having numerous ties and overlaps with various problems in random matrix models, probability, representation theory, and combinatorics. Much recent progress in these areas exploits the underlying notion of quantum integrability. Here we report on the scientific motivations and background of this activity and on its main outputs. [ABSTRACT FROM AUTHOR]

A tiling of a vector space S is the pair (U, V) of its subsets such that every vector in S is uniquely represented as the sum of a vector from U and a vector from V. A tiling is connected to a perfect codes if one of the sets, say U, is projective, i.e., the union of one-dimensional subspaces of S. A tiling (U, V) is full-rank if the affine span of each of U, V is S. For finite non-binary vector spaces of dimension at least 6 (at least 10), we construct full-rank tilings (U, V) with projective U (both U and V, respectively). In particular, that construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces. [ABSTRACT FROM AUTHOR]

The ability to engineer sophisticated two-dimensional tessellation organic nanoarchitectures based on triangular molecules and on-surface-synthesized covalent multimers is investigated using scanning tunneling microscopy. 1,3,5-Tris(3,5-dibromophenyl)benzene molecules are deposited on high-temperature Au(111) surfaces to trigger Ullmann coupling. The self-assembly into a semi-regular rhombitrihexagonal tiling superstructure not only depends on the synthesis of the required covalent building blocks but also depends on their ratio. The organic tessellation nanoarchitecture is achieved when the molecules are deposited on a Au(111) surface at 145 ° C . This halogen-bonded structure is composed of triangular domains of intact molecules separated by rectangular rows of covalent dimers. The nearly hexagonal vertices are composed of covalent multimers. The experimental observations reveal that the perfect semi-regular rhombitrihexagonal tiling cannot be engineered because it requires, in addition to the dimers and intact molecules, the synthesis of covalent hexagons. This building block is only observed above 165 ° C and does not coexist with the other required organic buildings blocks. [ABSTRACT FROM AUTHOR]

In this paper, we define new kinds of Fibonacci numbers, which generalize both Fibonacci, Jacobsthal, Narayana numbers and Fibonacci p-numbers in the distance sense, using the definition of a distance between numbers by a recurrence relation according to a new parameter k. Tiling and combinatorial interpretations of these numbers are presented, and explicit formulas that allow us to calculate the nth number are given. Also, their generating functions are obtained and sums formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties. [ABSTRACT FROM AUTHOR]

The present paper revisits the proof given in a paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is algorithmically unsolvable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is algorithmically unsolvable for the Euclidean plane, initially proved by Robert Berger in 1966. The present construction improves that of the 2008 paper. It also very strongly reduces the number of prototiles. [ABSTRACT FROM AUTHOR]

In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Pell, Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, Modified k-Pell, k-Jacobsthal numbers and Fibonacci p-numbers, distance Fibonacci numbers, (2,k)-distance Fibonacci numbers, generalized (k,r)-Fibonacci numbers in the distance sense by extending the definition of a distance in the recurrence relation with two parameters and adding three parameters in the definition of this distance, simultaneously. Tiling and combinatorial interpretations of generalized Fibonacci numbers are presented, and explicit formulas that allow us to calculate the nth number are given. Also generating functions and some identities for these numbers are obtained. [ABSTRACT FROM AUTHOR]

We define a class of subshifts defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. For such a subshift, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on convex sets. We show by example that these subshifts need not admit a group structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is "midpointed with respect to the shape". We construct such convex geometries on several groups, in particular strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, and show a link to conjectures of Gottshalk and Kaplansky. [ABSTRACT FROM AUTHOR]

The ability to engineer sophisticated two-dimensional tessellation organic nanoarchitectures based on triangular molecules and on-surface-synthesized covalent multimers is investigated using scanning tunneling microscopy. 1,3,5-Tris(3,5-dibromophenyl)benzene molecules are deposited on high-temperature Au(111) surfaces to trigger Ullmann coupling. The self-assembly into a semi-regular rhombitrihexagonal tiling superstructure not only depends on the synthesis of the required covalent building blocks but also depends on their ratio. The organic tessellation nanoarchitecture is achieved when the molecules are deposited on a Au(111) surface at 145 °C. This halogen-bonded structure is composed of triangular domains of intact molecules separated by rectangular rows of covalent dimers. The nearly hexagonal vertices are composed of covalent multimers. The experimental observations reveal that the perfect semi-regular rhombitrihexagonal tiling cannot be engineered because it requires, in addition to the dimers and intact molecules, the synthesis of covalent hexagons. This building block is only observed above 165 °C and does not coexist with the other required organic buildings blocks.

Consider a subshift over a finite alphabet, X ⊂ Λℤ (or X ⊂ Λℕ0). With each finite block B ∈ Λk appearing in X we associate the empirical measure ascribing to every block C ∈ Λl the frequency of occurrences of C in B .By comparing the values ascribed to blocks C we define a metric on the combined space of blocks B and probability measures μ on X, whose restriction to the space of measures is compatible with the weak-* topology. Next, in this combined metric space we fix an open set U containing all ergodic measures, and we say that a block B is "ergodic" if B ∈ U. In this paper we prove the following main result: Given ε > 0, every x ∈ X decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set M of coordinates of upper Banach density smaller than ε, all blocks in the decomposition are ergodic. We also prove a finitistic version of this theorem (about decomposition of long blocks), and a version about decomposition of x ∈ X into finite blocks of unbounded lengths. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how x ∈ X is partitioned into blocks (as long as their lengths are sufficiently large and bounded), excluding a set M of upper Banach density smaller than ε, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set M, in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts X ⊂ ΛG with the action of a countable amenable group G. The role of long blocks is played by blocks whose domains are members of a Følner sequence, while the decomposition of x ∈ X into blocks (of which majority are ergodic) is obtained with the help of a congruent system of tilings. [ABSTRACT FROM AUTHOR]

Given a finite set of local constraints, we seek a cellular automaton (i.e., a local and uniform algorithm) that self-stabilises on the configurations that satisfy these constraints. More precisely, starting from a finite perturbation of a valid configuration, the cellular automaton must eventually fall back into the space of valid configurations where it remains still. We allow the cellular automaton to use extra symbols, but in that case, the extra symbols can also appear in the initial finite perturbation. For several classes of local constraints (e.g., k-colourings with k ≠ 3, and North-East deterministic constraints), we provide efficient self-stabilising cellular automata with or without additional symbols that wash out finite perturbations in linear or quadratic time, but also show that there are examples of local constraints for which the self-stabilisation problem is inherently hard. We note that the optimal self-stabilisation speed is the same for all local constraints that are isomorphic to one another. We also consider probabilistic cellular automata rules and show that in some cases, the use of randomness simplifies the problem. In the deterministic case, we show that if finite perturbations are corrected in linear time, then the cellular automaton self-stabilises even starting from a random perturbation of a valid configuration, that is, when errors in the initial configuration occur independently with a sufficiently low density. [ABSTRACT FROM AUTHOR]

We investigate the internal observability of the wave equation with Dirichlet boundary conditions in tilings. The paper includes a general result relating internal observability problems in general domains to their tiles, and a discussion of the case in which the domain is the 30-60-90 triangle.

Quasicrystals have been discovered in a variety of materials ranging from metals to polymers. Yet, why and how they form is incompletely understood. In situ transmission electron microscopy of alloy quasicrystal formation in metals suggests an error-and-repair mechanism, whereby quasiperiodic crystals grow imperfectly with phason strain present, and only perfect themselves later into a high-quality quasicrystal with negligible phason strain. The growth mechanism has not been investigated for other types of quasicrystals, such as dendrimeric, polymeric, or colloidal quasicrystals. Soft-matter quasicrystals typically result from entropic, rather than energetic, interactions, and are not usually grown (either in laboratories or in silico) into large-volume quasicrystals. Consequently, it is unknown whether soft-matter quasicrystals form with the high degree of structural quality found in metal alloy quasicrystals. Here, we investigate the entropically driven growth of colloidal dodecagonal quasicrystals (DQCs) via computer simulation of systems of hard tetrahedra, which are simple models for anisotropic colloidal particles that form a quasicrystal. Using a pattern recognition algorithm applied to particle trajectories during DQC growth, we analyze phason strain to followthe evolution of quasiperiodic order. As in alloys, we observe high structural quality; DQCs with low phason strain crystallize directly from the melt and only require minimal further reduction of phason strain. We also observe transformation from a denser approximant to the DQC via continuous phason strain relaxation. Our results demonstrate that soft-matter quasicrystals dominated by entropy can be thermodynamically stable and grown with high structural quality--just like their alloy quasicrystal counterparts. [ABSTRACT FROM AUTHOR]

There is increasing interest in two-dimensional and quasi-two-dimensional materials and metamaterials for applications in chemistry, physics and engineering. Some of these applications are driven by the possible auxetic properties of such materials. Auxetic frameworks expand along one direction when subjected to a perpendicular stretching force. An equiauxetic framework has a unique mechanism of expansion (an equiauxetic mode) where the symmetry forces a Poisson’s ratio of −1. Hinged tilings offer opportunities for the design of auxetic and equiauxetic frameworks in 2D, and generic auxetic behaviour can often be detected using a symmetry extension of the scalar counting rule for mobility of periodic body-bar systems. Hinged frameworks based on Archimedean tilings of the plane are considered here. It is known that the regular hexagonal tiling, {63}, leads to an equiauxetic framework for both single-link and double-link connections between the tiles. For single-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found here to be equiauxetic: these are {3.122}, {4.6.12}, and {4.82}. For double-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found to be equiauxetic: these are {34.6}, {32.4.3.4}, and {3.6.3.6}.

Euclidean tilings are constantly applied to many fields of engineering (mechanical, civil, chemical, etc.). These tessellations are usually named after Cundy & Rollett’s notation. However, this notation has two main problems related to ambiguous conformation and uniqueness. This communication explains the GomJau-Hogg’s notation for generating all the regular, semi-regular (uniform) and demi-regular (k-uniform, up to at least k = 3) in a consistent, unique and unequivocal manner. Moreover, it presents Antwerp v3.0, a free online application, which is publicly shared to prove that all the basic tilings can be obtained directly from the GomJau-Hogg’s notation.

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]

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]

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 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]

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]

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]

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]

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.

We study the topological properties of a class of planar crystallographic replication tiles. Let M ∈ Z 2 × 2 be an expanding matrix with characteristic polynomial x 2 + A x + B ( A , B ∈ Z , B ≥ 2 ) and v ∈ Z 2 such that ( v , M v ) are linearly independent. Then the equation M T + B − 1 2 v = T ∪ ( T + v ) ∪ ( T + 2 v ) ∪ ⋯ ∪ ( T + ( B − 2 ) v ) ∪ ( − T ) defines a unique nonempty compact set T satisfying T o ¯ = T . Moreover, T tiles the plane by the crystallographic group p 2 generated by the π -rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T ∪ ( − T ) is homeomorphic to a closed disk if and only if 2 | A | < B + 3 . However, this characterization does not hold anymore for T itself. In this paper, we completely characterize the tiles T of this class that are homeomorphic to a closed disk. [ABSTRACT FROM AUTHOR]

Matt Blum conjectured that the- number of tilings of a hexagonal dungeon with side-lengths a , 2 a , b , a , 2 a , b (for b ≥ 2 a ) equals 1 3 2 a 2 1 4 ⌊ a 2 ∕ 2 ⌋ . Ciucu and the author of the present paper proved the conjecture by using Kuo’s graphical condensation method. In this paper, we investigate a 3-parameter refinement of the conjecture and its application to enumeration of tilings of several new types of the hexagonal dungeons. [ABSTRACT FROM AUTHOR]

In this paper, we present a construction of Kari–Culik aperiodic tileset. Our construction is self-contained and organized to allow reasoning on properties of the resulting sets of tilings. We prove that this tileset does not have any “unexpected behavior”, i.e., each line of each tiling has an average. Then we prove that this tileset has positive entropy, and that entropy is still positive when one adds some specific restrictions on the tilings. This shows that it is not self-similar, contrarily to all preceding aperiodic tilesets. [ABSTRACT FROM AUTHOR]

It is shown that there are primitive substitution tilings with dense tile orientations invariant under n -fold rotation for n ∈ { 2 , 3 , 4 , 5 , 6 , 8 } . The proof for dense tile orientations uses a general result about irrationality of angles in certain parallelograms. [ABSTRACT FROM AUTHOR]

A novel approach is proposed for the description of possible reconstructive solid-state transformations, which is based on the analysis of topological properties of atomic periodic nets and relations between their subnets and supernets. The concept of a region of solid-state reaction that is the free space confined by a tile of the net tiling is introduced. These regions (tiles) form the reaction zone around a given atom A thus unambiguously determining the neighboring atoms that can interact with A during the transformation. The reaction zone is independent of the geometry of the crystal structure and is determined only by topological properties of the tiles. The proposed approach enables one to drastically decrease the number of trial structures when modeling phase transitions in solid state or generating new crystal substances. All crystal structures which are topologically similar to a given structure can be found by the analysis of its topological vicinity in the configuration space. Our approach predicts amorphization of the phase after the transition as well as possible single-crystal-to-single-crystal transformations. This approach is applied to generate 72 new carbon allotropes from the initial experimentally determined crystalline carbon structures and to reveal four allotropes, whose hardness is close to diamond. Using the tiling model it is shown that three of them are structurally similar to other superhard carbon allotropes, M-carbon and W-carbon.

We study a problem of a number of lattice plane tilings by given area polyominoes. A polyomino is a connected plane geometric figure formed by joining edge to edge a finite number of unit squares. A tiling is a lattice tiling if each tile can be mapped to any other tile by translation which maps the whole tiling to itself. Let T(n) be a number of lattice plane tilings by given area polyominoes such that its translation lattice is a sublattice of Z². It is proved that 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. 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 polyomino and on the theory of self-avoiding walk. Also, it is proved that almost all polyominoes that give lattice plane tilings have sufficiently large perimeters.

In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure.

Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.

In this paper, we distinguish two levels for the plane-filling property. We consider a simple and a strong one. In this paper, we give the construction which proves that the simple plane-filling property also holds for the hyperbolic plane. The plane-filling property was established for the Euclidean plane by J. Kari, see [2], in the strong version. We also give an application of the construction to devise a Peano curve in the hyperbolic plane.

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.