Regular Tessellations Research Articles

On the zagreb connection indices of hex and honeycomb networks

Regular plane tessellations can easily be constructed by repeating regular polygons. This design is of extreme importance for direct interconnection networks as it yields high overall performance. The honeycomb and the hexagonal networks are two such popular mesh-derived parallel networks. The first and second Zagreb indices are among the most studied topological indices. We now consider analogous graph invariants, based on the second degrees of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for the Hex, Hex derived and some honeycomb networks.

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

Tilings of convex polygons by equilateral triangles of many different sizes

An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles (Tutte, 1948). Therefore Tuza (Tuza, 1991) asked for the largest number s=s(n) such that there is a tiling of an equilateral triangle by n equilateral triangles of s(n) different sizes. We solve that problem completely and consider the analogous questions for dissections of convex k-gons into equilateral triangles, k=4,5,6. Moreover, we discuss all these questions for the subclass of tilings such that no two tiles are translates of each other.

P.702 Feasibility of mobile dry-EEG for early detection of psychotic disorders: A usability study and pilot field trial

We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values for perimeter tiles in convex centrally symmetric 2n-gons. These bounds and values have implications for the combinatorics of reduced decompositions of permutations. We also classify the permutations whose polygons gave minimal perimeter, defined in two different ways. We conclude by looking at some of these questions in the context of domino tilings, giving a recursive formula and generating function for one family, and describing a family of minimal-perimeter regions.

Generic Green Skills for Creativity and Innovation: Tessellation of Regular Polygons

This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can freely be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.

Mechanical design and modelling of lightweight additively manufactured lattice structures evolved from regular three-dimensional tessellations

To lower costs and improve efficiency, most additively manufactured parts are printed as thin shells filled with a lightweight cellular material, which reacts to secondary stresses and provides distributed support to the load-carrying outer casing. This paper proposes a two-step method to design filling metamaterials that are intrinsically strong, stiff and lightweight. First, the space is divided into repetitive volumes according to available three-dimensional tessellation schemes. The tessellation is then transformed into a trabecular wireframe by converting each unit volume into a kinematically-rigid open cell with edge beams and bracings. Dimensional analysis allows the lattice structures to be characterized mechanically and elastically with a finite number of simple computational analyses. The paper shows that the properties of metamaterials with triangular, square and hexagonal prismatic cells compare favourably with state-of-the art porous materials like foams and honeycombs.

Multiscale SAR image segmentation by combining curvelet transform and GMTRJ algorithms

To integrate multiscale analysis within region-based segmentation framework, this paper presents multiscale SAR image segmentation method. First, curvelet transform is used to obtain a collection of the decomposed SAR images on multiple scales. Their domain is partitioned into a set of blocks by a regular tessellation, in which the number of blocks is assumed to be a random variable with Poisson distribution. On the partitioned domain, the Bayesian paradigm is followed to build the region- and multiscale-based image segmentation model with unknown number of classes. Further, a Generalized Multiple-Try Reversible Jump (GMTRJ) algorithm is designed to simulate the segmentation model. During the procedure of iterative simulation for the segmentation model, the segmentation result on the current scale is regarded as the initial segmentation on next scale. The segmented results corresponding to the finest scale is considered as the optimal image segmentation. The proposed method is validated using SAR images. The Kappa coefficient of the test simulated SAR image is up to 0.998, and the Kappa coefficients of the test real SAR images are up or equal to 0.903. From the test results of the quantitative and qualitative evaluation, it can be found that the proposed method can not only determine the number of classes, but also segment homogeneous regions well.

Tessellations in the Architecture of Pablo Palazuelo

After studying the unpublished sketches of the Spanish artist Pablo Palazuelo, this essay proposes the existence of geometric patterns that order his designs. To illustrate the proposal, this paper selects three representative projects by Palazuelo that attempt to demonstrate the presence of a working process related to a mathematical substrate. Specifically, the use of an irregular tiling is proposed for the creation of the following architectural elements: the scenery for a composition by Kandinsky (1950–1954), and the plan for a hotel on Princesa Street, Madrid, (1961). This study also examines the use of regular tessellations in the design of the ceilings of the Huarte Residence in Madrid, (1965). The treatises by Bourgoin and Ghyka have been selected from the library of Palazuelo as the primary theoretical bases of this study. This methodology constitutes a line of further research to analyze different architectural projects.

Supramolecular Tessellations by a Rigid Naphthalene Diimide Triangle.

Tessellation of organic polygons though [π···π] and charge-transfer (CT) interactions offers a unique opportunity to construct supramolecular organic electronic materials with 2D topologies. Our approach to exploring the 3D topology of 2D tessellations of a naphthalene diimide-based molecular triangle (NDI-Δ) reveals that the 2D molecular arrangement is sensitive to the identity of the solvent and solute concentrations. Utilization of nonhalogenated solvents, combined with careful tailoring of the concentrations, results in NDI-Δ self-assembling though [π···π] interactions into 2D honeycomb triangular and hexagonal tiling patterns. Cocrystallization of NDI-Δ with tetrathiafulvalene (TTF) leads systematically to the formation of 2D tessellations as a result of superstructure-directing CT interactions. Different solvents lead to different packing arrangements. Using MeCN, CHCl3, and CH2Cl2, we identified three sets of cocrystals, namely CT-A, CT-B, and CT-C, respectively. Solvent modulation plays a critical role in controlling not only the NDI-Δ:TTF stoichiometric ratios and the molecular arrangements in the crystal superstructures, but also prevents the inclusion of TTF guests inside the cavities of NDI-Δ. Confinement of TTF inside the NDI-Δ cavities in the CT-A superstructure enhances the CT character with the observation of a broad absorption band in the NIR region. In the CT-B superstructure, the CHCl3 lattice molecules establish a set of [Cl···Cl] and [Cl···S] intermolecular interactions, leading to the formation of a hexagonal grid of solvent in which NDI-Δ forms a triangular grid. In the CT-C superstructure, three TTF molecules self-assemble, forming a supramolecular isosceles triangle TTF-Δ, which tiles in a plane alongside the NDI-Δ, producing a 3 + 3 honeycomb tiling pattern of the two different polygons. Solid-state spectroscopic investigations on CT-C revealed the existence of an absorption band at 2500 nm, which on the basis of TDDFT calculations, was attributed to the mixed-valence character between two TTF•+ radical cations and one neutral TTF molecule.

TELPAC: A time and energy efficient protocol for locating and patching coverage holes in WSNs

Guaranteeing sufficient sensing coverage is of prime importance in wireless sensor networks. Unfortunately, due to many reasons such as natural disruptions, adversarial attacks, or energy depletion, the occurrence of coverage holes is unavoidable. In order to assure the quality of service, coverage holes should be patched (i.e. by deploying new sensors) as soon as they appear. The solutions in state-of-the-art protocols still incur time complexity and energy overhead that increase with the size of coverage holes. To avoid that issue, this paper introduces a novel protocol (namely, TELPAC) which efficiently locate the hole boundary and determine the patching locations. The main idea behind TELPAC is to approximate the hole by a polygon whose edges are aligned by a regular triangle lattice. Based on such approximation, the patching locations are then detected by using a regular hexagon tessellation. We theoretically prove that TELPAC can detect all coverage holes in the network. The simulation results show that the number of patching locations required by TELPAC is one of the smallest. Moreover, TELPAC can reduce more than 50% of the time consumed and energy overhead in comparison with the existing protocols.

Programming shape using kirigami tessellations.

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems

Orthogonal sequences can be assigned to a regular tessellation of hexagonal cells, typical for synchronised code-division multiple-access (S-CDMA) systems. In this paper, we first construct a new class of orthogonal sequences with increasing the number of users per cell to be \begin{document}$ 2^{m-2} $\end{document} for even number \begin{document}$ m\geq 4 $\end{document} (where \begin{document}$ 2^m $\end{document} is the length of the sequences). In addition, based on the above construction we construct a family of orthogonal sequences with zero correlation zone property which can be applied to the quasi-synchronous CDMA (QS-CMDA) spread spectrum systems.

The Perceived Beauty of Regular Polygon Tessellations

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.

On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean {3,122} semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE.

Semi-supervised map regionalization for categorical data

ABSTRACTThe objective of map regionalization is to group contiguous objects on a map into larger entities sharing similar properties or relationships, resulting in homogeneous regions that are easier to interpret. We propose a strategy to interactively incorporate human perception of homogeneous regions to improve unsupervised regionalization processes. The approach fits within the well-known segmentation/clustering framework. The method operates on a categorical map, introduces a contour detector for boundaries delineation with better resolution power than a regular grid tessellation to initiate a region growing process, and integrates the role of a human analyst for better classification of homogeneous areas through a semi-supervised clustering (SSC) method. This last step is achieved using pairwise clustering constraints on regions identified by the analyst on the monitor. The potential of the proposed strategy is illustrated with data extracted from the Earth Observation for the sustainable development of forests (EOSD) map of Canada. Comparisons with a recently introduced algorithm for map regionalization are provided for three different spatial scales at different steps of the method.

THE THE APPLICATION OF GENERIC GREEN SKILLS IN TESSELLATION OF REGULAR POLYGONS FOR ECONOMIC AND SOCIAL SUSTAINABILITY

Abstract
 This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can free be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.

Acquiring periodic tilings of regular polygons from images

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

An Energy-Based SAR Image Segmentation Method with Weighted Feature

To extract more structural features, which can contribute to segment a synthetic aperture radar (SAR) image accurately, and explore their roles in the segmentation procedure, this paper presents an energy-based SAR image segmentation method with weighted features. To precisely segment a SAR image, multiple structural features are incorporated into a block- and energy-based segmentation model in weighted way. In this paper, the multiple features of a pixel, involving spectral feature obtained from original SAR image, texture and boundary features extracted by a curvelet transform, form a feature vector. All the pixels’ feature vectors form a feature set of a SAR image. To automatically determine the roles of the multiple features in the segmentation procedure, weight variables are assigned to them. All the weight variables form a weight set. Then the image domain is partitioned into a set of blocks by regular tessellation. Afterwards, an energy function and a non-constrained Gibbs probability distribution are used to combine the feature and weight sets to build a block-based energy segmentation model with feature weighted on the partitioned image domain. Further, a reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm is designed to simulate from the segmentation model. In the RJMCMC algorithm, three move types were designed according to the segmentation model. Finally, the proposed method was tested on the SAR images, and the quantitative and qualitative results demonstrated its effectiveness.

A Coarse Model for the Multiaxial Elastic-Plastic Response of Ductile Porous Materials

We propose a modeling strategy to predict the mechanical response of porous solids to imposed multiaxial strain histories. A coarse representation of the microstructure of a porous material is obtained by subdividing a volume element into cubic cells by a regular tessellation; some of these cells are modeled as a plastically incompressible elastic-plastic solid, representing the parent material, while the remaining cells, representing the pores, are treated as a weak and soft compressible solid displaying densification behavior at large compressive strains. The evolution of homogenized deviatoric and hydrostatic stress is explored for different porosities by finite element simulations. The predictions are found in good agreement with previously published numerical studies in which the microstructural geometry was explicitly modeled.

Golden codes: quantum LDPC codes from regular tessellations of hyperbolic 4-manifolds

We adapt a construction of Guth and Lubotzky \cite{GL14} to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like n^{0.1} where n is the number of physical qubits. Similarly as in Ref.~\cite{GL14}, our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.