Search

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

Given graphs F and G, a perfect F-tiling in G is a collection of vertex-disjoint copies of F in G that together cover all the vertices in G. The study of the minimum degree threshold forcing a perfect F-tiling in a graph G has a long history, culminating in the Kühn--Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65-107] which resolves this problem, up to an additive constant, for all graphs F. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs F this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect P-tiling in an edge-ordered graph, where P is any fixed monotone path. [ABSTRACT FROM AUTHOR]

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]

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]

A polygon P is called a reptile, if it can be decomposed into k ≥ 2 nonoverlapping and congruent polygons similar to P. We prove that if a cyclic quadrilateral is a reptile, then it is a trapezoid. Comparing with results of Betke and Osburg we find that every convex reptile is a triangle or a trapezoid. [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]

We present Planar Rosa, a family of rhombus tilings with a 2n-fold rotational symmetry that are generated by a primitive substitution and that are also discrete plane tilings, meaning that they are obtained as a projection of a higher dimensional discrete plane. The discrete plane condition is a relaxed version of the cut-and-project condition. We also prove that the Sub Rosa substitution tilings with 2n-fold rotational symmetry defined by Kari and Rissanen do not satisfy even the weaker discrete plane condition. We prove these results for all even n ⩾ 4 . This completes our previously published results for odd values of n. [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]

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]

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. [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]

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 give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For p = 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph's boundary in terms of square tilings of cylinders. [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]

This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$. By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin. [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]

Textile pattern design is an important component of textile production. Patterns with artistic beauty can inject added value into textiles and improve market competitiveness. The traditional way of designing patterns mainly relies on manuals which requires designers to explore creative inspiration in a short period of time and thus is time consuming and laborious. Therefore it is sometimes difficult to meet the personalized demands in the current "fast-paced" society. The problem can be partially solved by using digital art graphics which are based on mathematical models e. g. fractal graphics dynamical systems graphics quasi-regular patterns etc. With the help of computer technology a large number of digital art graphics with natural beauty and rich changes can be generated very efficiently. Such graphics can be used as textile pattern materials for secondary design. Thus it can significantly improve the design efficiency and provide new creative inspiration for traditional textile pattern design. However most existing methods usually generate patterns tiling the whole image space which is difficult to extract single patterns as designing material. Some methods can generate single patterns but are hard to control its variations. To this end we propose a single pattern generation method based on spherical harmonic functions and construct conditions for generating patterns with symmetric property. First the mesh surface is sampled and discretized from a specific closed-form spherical harmonic function with several parameters. Then the surface is segmented into several closed domains and colored separately by contour segmentation. And the 2D pattern can be obtained by vertical projection. To control the pattern style different types of symmetry conditions for parameters are constructed. Compared with those methods tiling the whole image space patterns generated by our method have single independent structures and can be directly used as the basic materials for design and save the efforts of element extraction. Besides our method is much easier to control the variation of the pattern by adjusting parameters with symmetric conditions. Finally we apply the generated graphics combined with the tiling structure to design textile patterns and simulate products as silk scarf pillow and carpet. In the future more control techniques and vectorization techniques will be further studied for the pattern generation so as to provide a more convenient means for designers to carry out secondary design. [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]

We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike Z 2 , but like Z 3 , strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on B S (m , n) due to Aubrun and Kari is, in fact, strongly aperiodic on B S (1 , n) ; and weakly but not strongly aperiodic on any other B S (m , n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on B S (1 , n) ; and we show that there exists a strongly aperiodic SFT on B S (n , n). [ABSTRACT FROM AUTHOR]

In this paper, I try to explain how, by using concepts and ideas from the mathematical theory of tilings, we can approach metre in music through a geometric and algebraic point of view, being pinned down by a subgroup of R with the hierarchical structure, leading to an abstract approach to rhythm, tempo and time signatures. I will also describe an algorithmic approach to write down sound using this structure which gives a way in which music can be written in an irrational metre. [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 examine the aesthetic characteristics of row tile patterns defined by repeating strips of polygons. In experiment 1 participants rated the perceived beauty of equilateral triangle, square and rectangular tilings presented at vertical and horizontal orientations. The tiles were shifted by one-fourth increments of a complete row cycle. Shifts that preserved global symmetry were liked the most. Local symmetry by itself did not predict ratings but tilings with a greater number of emergent features did. In a second experiment we presented row tiles using all types of three- and four-sided geometric figures: acute, obtuse, isosceles and right triangles, kites, parallelograms, a rhombus, trapezoid, and trapezium. Once again, local polygon symmetry did not predict responding but measures of correspondence between local and global levels did. In particular, number of aligned polygon symmetry axes and number of aligned polygon sides were significantly and positively correlated with beauty ratings. Preference was greater for more integrated tilings, possibly because they encourage the formation of gestalts and exploration within and across levels of spatial scale. [ABSTRACT FROM AUTHOR]

We define a notion of tiling of the full infinite p-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and topological. We show how these results can be applied in a straightforward and constructive manner to define homeomorphisms between two given spaces of p-adic integers, Z p and Z q , endowed with their corresponding standard non-archimedean metric topologies. [ABSTRACT FROM AUTHOR]

A tiling is k − isotoxal if its edges form k orbits or k transitivity classes under the action of its symmetry group. In this article, a method is presented that facilitates the systematic derivation of planar edge-to-edge k − isotoxal tilings from isohedral [ p n ] tilings. Two well-known subgroups of triangle groups will be used to create and determine classes of k − isotoxal tilings in the Euclidean, hyperbolic and spherical planes which will be described in terms of their symmetry groups and symbols. The symmetry properties of k − isotoxal tilings make these appropriate tools to create geometrically influenced artwork such as Escher-like patterns or aesthetically pleasing designs in the three classical geometries. [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.

Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift Ω 0 made of all valid tilings using the set T 0 of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift X 0 of Ω 0 such that every tiling in X 0 can be decomposed uniquely into 19 distinct patches of size ranging from 45 to 112 that are equivalent to a set of 19 self-similar aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel–Rao tilings, as we believe that Ω 0 \ X 0 is a null set for any shift-invariant probability measure on Ω 0 . The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles, while the other 2 involve addition decorations to deal with fault lines and changing the base of the Z 2 -action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles. [ABSTRACT FROM AUTHOR]

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. [ABSTRACT FROM AUTHOR]

This paper establishes a fundamental difference between $\mathbb{Z}$ subshifts of finite type and $\mathbb{Z}^{2}$ subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type $X$ as a subset of a full shift  $F$. We then introduce a natural penalty function  $f$ , defined on  $F$ , which is 0 if the local configuration near the origin is legal and $-1$ otherwise. We show that in the case of $\mathbb{Z}$ subshifts, for all sufficiently small perturbations, $g$ , of  $f$ , the $g$ -maximizing invariant probability measures are supported on $X$ (that is, the set $X$ is stably maximized by  $f$). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations, $g$ , of  $f$ for which the $g$ -maximizing invariant probability measures are supported on $F\setminus X$. [ABSTRACT FROM AUTHOR]

In this article, we calculate the Ellis semigroup of a certain class of constant-length substitutions. This generalizes a result of Haddad and Johnson [IP cluster points, idempotents, and recurrent sequences. Topology Proc.22 (1997) 213–226] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counterexamples to one of the propositions in their paper, which is central to the proof of their main theorem. We give an alternative approach to their result, which centers on the properties of the Ellis semigroup. To do this, we also show a new way to construct an almost automorphic–isometric tower to the maximal equicontinuous factor of these systems, which gives a more particular approach than the one given by Dekking [The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheor. Verw. Geb.41(3) (1977/78) 221–239]. [ABSTRACT FROM AUTHOR]

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a Pisot–Vijayaraghavan (PV) number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We discuss the connection between a discrete tiling of a lattice and a continuous tilling of the n-dimensional space and equivalent characterizations of discrete tilings. We generalize bounds on the radius of perfect codes in a generic lattice, which were previously known for the cubic lattice, and we provide some new bounds. The new bounds are based on the packing and the covering densities and on the covering radius of the ambient lattice. An algorithm is presented for the search of perfect codes which is used to derive all perfect codes for a collection of ambient lattices in dimensions two and three. In contrast to the cubic lattice, these case studies show that by considering general ambient lattices one can find rich sets of perfect codes. [ABSTRACT FROM AUTHOR]

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]

Copyright of KoG is the property of Croatian Society for Geometry & Graphics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This review of mathematical art exhibited at Bridges Linz in 2019 focusses on the use of two 20th c. mathematical developments, the theory of tilings and aperiodic order, and of fractals, in the work of Dugan Hammock, Ryan Webb, Lisa Shier, Doug Dunham, Pallavi Nataragan, Peter Stempfli, Saara Lehto, Andrea Dozmati, Hans Kuiper, Bernat Espigulé, Mingjang Chen, Flora Olivier, Kevin Lee and Demian Nahuel Goos Bosco. [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.