Regular Tilings Research Articles

Fibre-Like Cylinders, Their Packings and Coverings in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} Space

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups pq21\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {pq2_1}$$\\end{document} we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in H3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^3$$\\end{document} and H2×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^2\\!\ imes \\!\ extbf{R}$$\\end{document} spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

A new approach for configuring modular floating cities: assessing modular floating platforms by means of analytic hierarchy process

Floating cities have emerged as an efficient long-term solution over unsustainable practiced solutions to combat the rising seas problem; nevertheless, the world lacks an international, official, and comprehensive framework regarding floating cities. Although previous research approached modular floating city design; however, resulted in configurations with various critical design restrictions mainly regarding interlocking capabilities and space utilization. The purpose of this paper is to offer a new systematic strategy for configuring modular and expandable floating cities without such restrictions. This paper explores Euclidean tilings as a strategy to offer numerous configurations based on regular, semi-regular, and demi-regular tilings. Selecting the ideal configuration is complicated; therefore, both quantitative and qualitative data methods were implemented to attain the objectives. Via an extensive literature review, this research derives key factors for configuring floating cities, then sets a brainstorming session with experts for group decision making before providing findings upon calculations via analytic hierarchy process, one of the most used quantitative data methods of multiple-criteria decision analysis. Through comprehensive literature review: seakeeping, modularity, zoning and circulation, and feasibility have been identified as the most significant criteria in floating city research. It explores the qualities and limitations of triangular, squared, hexagonal, octagonal, and dodecagonal platforms. Regarding criteria, seakeeping was the most significant criterion for platform selection by 53.6%. Regarding platforms, the hexagonal platform scored the highest with 25.31%. Relying on this method and the design considerations presented, numerous dynamic configurations can be offered and assessed through specific contexts without any of the past restrictions.

Stratified sample tiling

The paper introduces a practical method for the construction of large-scale point sets for analysis of computer models. The constructed experimental design is useful for (probabilistic) integration, construction of approximation or a screening. The essence of the presented approach is the stratification of the design domain into an orthogonal grid of substrata and a subsequent tiling with tiles of points. If optimized, such tiles experience a major reduction of the number of degrees of freedom in the optimization process. That way, optimal or near-optimal point patterns can be feasibly identified and are further utilized for construction of larger point sets, thanks to the idea of self-similarity and structured space stratification The space-filling properties of the resulting point sets may be further enhanced by various “scrambling” strategies, which may remove the undesired sample collapsibility achieved via regular tiling. The performance of the constructed point sets is compared to Quasi Monte Carlo (QMC), Randomized Quasi Monte Carlo (RQMC) sequences, which are still today considered by engineers and even scientists as choices for variance reduction of numerical integration Further, the mentioned sampling strategies are compared in the terms of robustness when integrating a multivariate function with a localized feature. It is concluded that the proposed sampling approach reaches a superior performance in numerical integration and identification of function extremes as compared to sampling methods used by practicing researchers and engineers. Additionally, the reader is supplied with the open-access, ready-to-use implementation of the presented algorithm named SampleTiler.

Constructing and Visualizing Uniform Tilings

This paper describes a system which takes user input of a pattern of regular polygons around one vertex and attempts to construct a uniform tiling with the same pattern at every vertex by adding one polygon at a time. The system constructs spherical, planar, or hyperbolic tilings when the sum of the interior angles of the user-specified regular polygons is respectively less than, equal to, or greater than 360∘. Other works have catalogued uniform tilings in tables and/or illustrations. In contrast, this system was developed as an interactive educational tool for people to learn about symmetry and tilings by trial and error through proposing potential vertex patterns and investigating whether they work. Users can watch the rest of the polygons being automatically added one by one with recursive backtracking. When a trial polygon addition is found to violate the conditions of a regular tiling, polygons are removed one by one until a configuration with another compatible choice is found, and that choice is tried next.

On the perimeter estimation of pixelated excursion sets of two‐dimensional anisotropic random fields

AbstractWe are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover , the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite‐sample data setting.

Breitenlohner-Freedman Bound on Hyperbolic Tilings.

We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass squared m^{2}. This follows from a real and positive total energy of the gravitational system. For finite cutoff ϵ, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When ϵ→0, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows us to further scan values of m^{2} above the BF bound.

Recovering cyclic tilings through β-skeletons

There is a class of proximity graphs parameterized by a real number β that are called β-skeletons and which are geometrical graphs whose vertex set V is a discrete set of points in the plane and two points p and q are adjacent if certain region luneβ(p, q) defined by them does not contain any other vertex. Special cases of β-skeletons are the Gabriel Graph (β = 1) and the Relative Neighborhood Graph (β = 2). Cyclic tilings are tilings of the plane whose tiles are convex cyclic polygons that contain the center of the circle in which they are inscribed. Up to our knowledge, this class of tilings has not been studied before. They generalize regular, Archimedean, and k-uniform tilings. We give sufficient conditions for the graph of a cyclic tiling to be a β-skeleton of its vertices. The result is particularly nice for the case of the Gabriel graph. The study of the relationship between tilings arises while studying street networks. This relationship is illustrated briefly.

Water model for hydrophobic cavities: structure and energy from quantum-chemical calculations.

This ab initio study aims to design a series of large water clusters having a hollow clathrate-like cage able to host hydrophobic solutes of various sizes. Starting from the (H2O)n (n = 18, 20, 24 and 28) hollow cages, water layers have been added in a stepwise manner in order to model the configuration of water molecules beyond the primary shell. The large (H2O)100, (H2O)120 and (H2O)140 clusters complete the hydrogen bonding network of the cage with optimal and regular tiling of the do-, tetra-decahedron and hexa-decahedron, respectively. This study is corroborated by an investigation of dense water clusters up to the (H2O)123 one, being highly consistent with experimental data on ice concerning the electronic and zero-point energies for aggregate formation at 0 K and enthalpy and entropy at 273 K. The cavity creation profoundly alters the orientation of water molecules compared with those found in dense clusters. Nevertheless, such a large reorganization is necessary to maximize the water-water attraction by making it similar to the one found in dense clusters. The cage formation is an endothermic process; however, the computed values are large compared with previous reports for hydrocarbon aqueous solutions. Larger clusters are required for a more fruitful comparison.

Design of Aperiodic Subarrayed Phased Arrays With Structural Repetitiveness

To eliminate the grating lobes of subarrayed phased arrays, the state-of-the-art methods mainly use subarrays of various shapes to make up an irregular subarray tiling configuration (irRSTC), so as to break the periodicity of the subarray tiling configuration. However, it leads to high design complexity due to the lack of structural repetitiveness. Different from the previous idea, the periodicity of the subarray tiling configuration is broken by introducing asymmetrical feeding phase into subarrays in this article. Specifically, a regular subarray tiling configuration (RSTC) is proposed, and it is obtained by replication and flipping of a regular subarray with asymmetrical feeding phase. In this way, the structural repetitiveness is achieved, while the periodicity of the RSTC is broken. Furthermore, to enhance the beam scanning ability, the aperiodicity of the RSTC is maximized by minimizing the predictability of the RSTC, based on the anticorrelation relationship between the aperiodicity and the predictability. Numerical results show that compared with the irRSTC, the proposed RSTC can expand the beam scanning range by 10° and improve the directivity of larger angle scanning by 2.9 dB only at an expense of 1.9 dB reduction at the boresight. Besides, the design complexity of the proposed RSTC is reduced to 1/64 of that of the irRSTC for a 16 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times16$ </tex-math></inline-formula> array.

Aharonov-Bohm cages, flat bands, and gap labeling in hyperbolic tilings

Aharonov-Bohm caging is a localization mechanism stemming from the competition between the geometry and the magnetic field. Originally described for a tight-binding model in the dice lattice, this destructive interference phenomenon prevents any wavepacket spreading away from a strictly confined region. Accordingly, for the peculiar values of the field responsible for this effect, the energy spectrum consists of a discrete set of highly degenerate flat bands. In the present work, we show that Aharonov-Bohm cages are also found in an infinite set of hyperbolic dice tilings defined on a negatively curved hyperbolic plane. We detail the construction of these tilings and compute their Hofstadter butterflies by considering periodic boundary conditions on high-genus surfaces. As recently observed for some regular hyperbolic tilings, these butterflies do not manifest the self-similar structure of their Euclidean counterparts but still contain some gaps. We also consider the energy spectrum of hyperbolic kagome tilings (which are the dual of hyperbolic dice tilings), which displays interesting features, such as highly degenerate states arising for some particular values of the magnetic field. For these two families of hyperbolic tilings, we compute the Chern number in the main gaps of the Hofstadter butterfly and propose a gap labeling inspired by the Euclidean case. Finally, we also study the triangular Husimi cactus, which is a limiting case in the family of hyperbolic kagome tilings, and we derive an exact expression for its spectrum versus magnetic flux.

Mechanical Characterization of Compliant Cellular Robots. Part I: Passive Stiffness

Abstract Modular active cell robots (MACROs) are a design paradigm for modular robotic hardware that uses only two components, namely actuators and passive compliant joints. Under the MACRO approach, a large number of actuators and joints are connected to create mesh-like cellular robotic structures that can be actuated to achieve large deformation and shape change. In this two-part paper, we study the importance of different possible mesh topologies within the MACRO framework. Regular and semi-regular tilings of the plane are used as the candidate mesh topologies and simulated using finite element analysis (FEA). In Part 1, we use FEA to evaluate their passive stiffness characteristics. Using a strain-energy method, the homogenized material properties (Young's modulus, shear modulus, and Poisson's ratio) of the different mesh topologies are computed and compared. The results show that the stiffnesses increase with increasing nodal connectivity and that stretching-dominated topologies have higher stiffness compared to bending-dominated ones. We also investigate the role of relative actuator-node stiffness on the overall mesh characteristics. This analysis shows that the stiffness of stretching-dominated topologies scales directly with their cross-section area whereas bending-dominated ones do not have such a direct relationship.

Universality of Hofstadter Butterflies on Hyperbolic Lattices.

Motivated by recent realizations of hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. Utilizing large hyperbolic lattices with periodic boundary conditions, we obtain the true bulk spectrum unaffected by boundary states. The butterfly spectrum with large extended gapped regions prevails, and its shape is universally determined by the fundamental tile, while the fractal structure is lost. We explain how these features originate from Landau levels in hyperbolic space and can be verified experimentally.

Boundary theories of critical matchgate tensor networks

Key aspects of the AdS/CFT correspondence can be captured in terms of tensor network models on hyperbolic lattices. For tensors fulfilling the matchgate constraint, these have previously been shown to produce disordered boundary states whose site-averaged ground state properties match the translation-invariant critical Ising model. In this work, we substantially sharpen this relationship by deriving disordered local Hamiltonians generalizing the critical Ising model whose ground and low-energy excited states are accurately represented by the matchgate ansatz without any averaging. We show that these Hamiltonians exhibit multi-scale quasiperiodic symmetries captured by an analytical toy model based on layers of the hyperbolic lattice, breaking the conformal symmetries of the critical Ising model in a controlled manner. We provide a direct identification of correlation functions of ground and low-energy excited states between the disordered and translation-invariant models and give numerical evidence that the former approaches the latter in the large bond dimension limit. This establishes tensor networks on regular hyperbolic tilings as an effective tool for the study of conformal field theories. Furthermore, our numerical probes of the bulk parameters corresponding to boundary excited states constitute a first step towards a tensor network bulk-boundary dictionary between regular hyperbolic geometries and critical boundary states.

Equiauxetic Hinged Archimedean Tilings

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}.

Cholesteric Shells: Two-Dimensional Blue Fog and Finite Quasicrystals.

We study the phase behavior of a quasi-two-dimensional cholesteric liquid crystal shell. We characterize the topological phases arising close to the isotropic-cholesteric transition and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two-dimensional topological phases: finite quasicrystals and amorphous structures, both made up of mixtures of polygonal tessellations of half-skyrmions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilizes heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals and amorphous and heterogeneous structures could be sought experimentally by self-assembling cholesteric shells on the surface of emulsion droplets.

Geodesic ball packings generated by regular prism tilings in Nil geometry

Geodesic ball packings generated by regular prism tilings in Nil geometry

Noise attenuating performance of metasurfaces with regular Euclidean tiling and uniformly embedded Helmholtz resonators

Sound-attenuation capabilities of a certain class of subwavelength sound barriers based on wall-embedded Helmholtz resonators and a ventilation duct are studied theoretically and numerically. A simple analytical expression for the broadband integral transmission attenuation is introduced for this type of metasurface structures, which indicates that the sound blocking performance depends only on the main operational frequency, thickness and ventilation capacity of the structure. This result derived from the lumped parameter theory implies that the trade-off between the barrier performance and its physical footprint is inevitable and provides a practical guidance for the design strategies with specific targets and limitations. Detailed finite element modelling (FEM) parameter study is performed both to validate the theoretical predictions and test the limitations of simple lumped parameter approach. It was found that the agreement between the analytical and finite element results is very good for the structure thickness greater than 30 mm in a 300–1000 Hz frequency range. However, at smaller thickness values the FE results start to diverge from the predictions and also become dependent on the specific cell shape, which suggests that more advanced analytical approach is required for thinner metasurfaces.

Polynomial-reproducing spline spaces from fine zonotopal tilings

Given a point configuration A, we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the dual graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions.

Firefighting on the hexagonal grid

The firefighter problem with k firefighters on an infinite graph G is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex v∈V(G) on Turn 1. On each subsequent even turn, k firefighters protect k vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters’ goal is to eventually stop the spread of the fire. If there exists a strategy for k firefighters to eventually stop the spread of the fire, then we say G is k-containable.We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is 1-containable. In Gavenčiak et al. (2014), it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire.

Cylindrical Super‐Lattice Structures with Three‐Contrasts from Pentablock Binary Blends Studied by Monte Carlo Simulation

AbstractThe cylindrical super‐lattice structures with three‐contrasts from binary blends of four‐component pentablock quadpolymers, AB0CB1D and CB0AB1D, is studied, where B0 and B1 are the same polymer component. The difference in the architecture between two quadpolymers consists merely in the order of A and C block chains. The volume fractions of the first‐, third‐, and fifth‐components are denoted as φ1, φ3, and φ5, respectively, and γ is the volume fraction ratio of two B‐chains, γ = φB0/φB1. To form cylindrical domains with three‐contrasts in B‐matrix phase, φ1, φ3, and φ5 are fixed as φ1 = φ5 = 0.125 and φ3 = 0.156. At the large γ region, the (2+2) and (3+3) tiling super‐lattice structures appear, where each D‐domain is surrounded with two or three A‐ and C‐cylinders each and their basic tilings are of squares and regular hexagons. At the smaller γ region, untraditional (4+4) and (5+5)(3) cylindrical super‐lattice structures are observed, where each D‐domain is surrounded with four or five A‐ and C‐cylinders each as satellites. Since square and regular hexagon can give Euclidean regular tilings, traditional (2+2) and (3+3) structures can be easily formed, while (4+4) and (5+5)(3) untraditional cylindrical super‐lattice structures are exhibited simply because regular octagons and decagons cannot fill out 2D surface.