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The article "Mathematics Logbook" describes the discovery of new mathematical tile shapes that include tessellations with rounded corners. The work "Soft cells and the geometry of seashells" by Domokos, Goriely, Horváth, and Regös describes these new tessellations that occur in biology. It also discusses Alhambra patterns and hyperbolic tessellations. The discovery of quasicrystals and aperiodic partitions is also discussed. [Extracted from the article]

A new coordination polymer (CP) with composition [Zn3(bpyp)2(TCB)2(H2O)2]n (1) has been engendered by performing reaction of Zn(II) with 1,3,5‐tris‐(4‐carboxyphenyl)benzene (H3TCB) and 2,5‐bis(pyrid‐4‐yl)pyridine (bpyp) under solvothermal condition. The single crystal X‐ray diffraction analysis for 1 suggests kgd topological behavior having resemblance with a tessellation composed of diamond shapes arranged in a star pattern and represented by the Schläfli symbol {41}2{42·62·81}. This newly designed CP exhibits optical band gap of 2.92 eV and employed as photocatalyst for photodegradation of antibiotics, namely, nitrofurazone (NFZ), chloramphenicol (CAP), sulfasalazine (SLA), oxytetracycline (OXY), and furazolidone (FZD). Among these antibiotics, 1 displays best photocatalytic activity against the photodegradation of NFZ and under optimized conditions viz. photocatalyst dosage of 15 mg, and initial NFZ concentration of 40 ppm, the CP 1 displays best catalytic performance and photo‐decomposed 93.04% NFZ with pseudo first order rate constant k of 0.04502 min−1 in 1 h. The likely photocatalytic mechanism has been evaluated using theoretical calculations. [ABSTRACT FROM AUTHOR]

The simplification operation has always been a challenging research problem in the field of cartography generalization and expression. Continuous tessellation area data of land cover, whose source data come from high-resolution remote sensing satellite sensors, are an important type of remote sensing application data to express the spatial distribution information of natural and artificial objects on Earth's surface. In this paper, a novel and practical simplification method for continuous tessellation area data is proposed. The first step is area discretization, in which a general topological structure containing semantic features is established, and land patches are divided into multiple arcs on the basis of this topological structure. The second step is arc simplification, in which arcs are classified into artificial and natural types according to their semantic and morphological features, and corresponding simplification algorithms are assigned. The final step is area reconstruction, based on the semantic and topological information, and simplified arcs are reconstructed to patch polygons. The geography and national conditions census data of a city in Guizhou Province, China, are used to verify the rationality and effectiveness of the method. Experimental results showed that the method proposed in this article is suitable for the simplification processing of artificial and natural land features. The simplification results can maintain the overall shape of the patches and reduce the complexity of the arcs effectively, and the semantic information of each patch after simplification is also correct. [ABSTRACT FROM AUTHOR]

This paper introduces a topology optimization approach based on the virtual element method (VEM), incorporating uncertainties. The objective of this optimization process is to design an optimal material layout for problems governed by linear elasticity equations to minimize the volume while satisfying probabilistic compliance constraints. The VEM is used to solve the boundary value problem in reliability-based topology optimization (RBTO). In the comparison between the VEM and the standard finite element method (FEM), a key difference emerges in the absence of explicitly defined shape functions tied to discrete degrees of freedom in VEM. Unlike FEM, VEM directly constructs the discrete bilinear form and load linear form without the need for computing shape function derivatives within the elements. This flexibility accommodates meshes with intricate geometries and arbitrarily shaped elements. The paper discusses the computational efficiency of VEM RBTO and explores the geometric impact of tessellations on converged topologies, demonstrating reduced susceptibility to checkerboard patterns compared to conventional quadrilateral elements. Additionally, the single-loop approach is examined, showcasing comparable accuracy to the first-order/second-order reliability methods (FORM/SORM) of RBTO using VEM. Numerical results for several problems that demonstrate the feasibility of the proposed method are presented. [ABSTRACT FROM AUTHOR]

Motivated by heat dissipation, the rigid-compliant hybrid cellular expansion mechanisms with motion amplification and superposition are proposed in this paper. Compared with existing studies, the expansion mechanism is not only easy to realize the plane tessellation via cellular design due to its regular polygon structure but also has the ability of motion amplification and superposition due to its compliant displacement amplifier and rigid scissors. First, the scheme of expansion mechanisms, especially the working principle of motion amplification and superposition, is introduced. The configuration design of a family of expansion mechanisms is presented, including varying number of edges, concave/convex property, and inner/outer layout. Second, the constraint condition and analytical modeling of relations between output performances of expansion mechanisms and dimensional parameters are carried out. Third, the displacement amplification ratio of expansion mechanisms and output performances of several typical expansion mechanisms when they act as cells to tessellate a plane with a constrained area are analyzed. Finally, the output performances of expansion mechanisms are verified via the finite element analysis. The results show that proposed cellular expansion mechanisms are beneficial for realizing plane tessellation and offer motion amplification and superposition, which provide prospects in the field of mechanism design such as metamaterials. [ABSTRACT FROM AUTHOR]

A new family of scalable codes with locality and availability for information repair in data storage systems for e-health applications was presented recently. The construction was based on a graph of the [7, 3-4] Simplex code. In this paper it is shown that the construction can be generalized via tessellation in a Euclidian plane. The codes obtained have new interesting recoverability properties. They can in some cases repair damage to many storage nodes in multiple connected graphs via sequential decoding, which is similar to healing wounds in biological systems. The advantages of the original codes, namely the availability, functionality, efficiency and high data accessibility, will be preserved also in these new codes. The computational complexity and communication costs of their incrementation will remain constant and modest. These codes could be adapted to disaster recovery because it is straightforward to place the nodes so that the graph is easily mapped on a real structure in space. [ABSTRACT FROM AUTHOR]

Processes of random tessellations of the Euclidean space $\mathbb{R}^d$ , $d\geq 1$ , are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matemática Española is the property of Real Sociedad Matematica Espanola 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.)

We trace the connectivity of the cosmic web as defined by haloes in the Planck-Millennium simulation using a persistence and Betti curve analysis. We normalize clustering up to the second-order correlation function and use our systematic topological analysis to correlate local information and properties of haloes with their multiscale geometrical environment of the cosmic web (elongated filamentary bridges and sheetlike walls). We capture the multiscale topology traced by the halo distribution through filtrations of the corresponding Delaunay tessellation. The resulting nested alpha shapes are sensitive to the local density, perfectly outline the local geometry, and contain the complete information on the multiscale topology. We find a remarkable linear relationship between halo masses and topology: haloes of different mass trace environments with different topological signature. This is a topological bias , an environmental structure bias independent of the halo clustering bias associated with the two-point correlation function. This mass-dependent linear scaling relation allows us to take clustering into account and determine the overall connectivity from a limited sample of galaxies. The presence of topological bias has major implications for the study of voids and filaments in the observed distribution of galaxies. The (infra)structure and shape of these key cosmic web components will strongly depend on the underlying galaxy sample. Their use as cosmological probes, with their properties influenced by cosmological parameters, will have to account for the subtleties of topological bias. This is of particular relevance with the large upcoming galaxy surveys such as DESI, Euclid, and the Vera Rubin telescope surveys. [ABSTRACT FROM AUTHOR]

This work presents an analytical model to predict the electrical conductivity of hybrid nanocomposites consisting of conducting nanowires and insulating particles. The model utilizes a single variable, the ratio of particle diameter to nanowire length, determined through data mining techniques. Machine learning techniques show a monotonically increasing relationship between this variable and conductivity. The conductivity equation is derived from the Kozeny-Carman relation, which relates conductivity to nanowire density and current path tortuosity. The Voronoi tessellation simplifies the filler distribution, allowing analytic expressions for these two characteristics. Results are compared to numerical simulation data to validate the model's accuracy. Highlights • Novel equation developed for nanocomposite design based on physical parameters. • Data mining identifies key parameters governing the conductivity of nanocomposites. • Machine learning reveals the relationship between key parameters and conductivity. • Computational geometry quantizes the dispersion structure of fillers. • Derivation of a conductivity equation using the Kozeny-Carman relation. [ABSTRACT FROM AUTHOR]

Voronoi treemaps are widely used for hierarchical data visualization. Existing methods calculate the visualization layouts of hierarchical data by combining the proportion optimization of weights and Lloyd's method of sites. However, this may not only produce results with large area errors but also require more time consumption. Besides, the relative visualization position of the same data element between adjacent frames in dynamic hierarchical data may be changed abruptly, resulting in unclear visual results. To this end, we propose an efficient and topological structure preserved visualization approach, called PowerHierarchy, for visualizing hierarchical data. Firstly, an improved version of the power diagram computing algorithm is introduced to generate the visualization layouts of each data element in the hierarchy. Unlike random initialization, we construct a centroidal Voronoi tessellation as input and then use a Breadth-First traversing strategy to adapt the depth information to produce visual layouts of static hierarchical data. Based on this, an updating scheme is presented for visualizing dynamic hierarchical data, where previous results are iteratively fed as inputs to initialize current layouts. Besides, the external boundary sites and their subsites are projected onto the visual boundary and then moved into the visual region with the relative position preserved. Experimental results on several datasets demonstrate the efficiency, accuracy, and topology preservation advantage of our proposed visualization approach. [ABSTRACT FROM AUTHOR]

Stationary Poisson processes of lines in the plane are studied, whose directional distributions are concentrated on $k\geq 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles's classical result for the isotropic case is obtained by an approximation argument. [ABSTRACT FROM AUTHOR]

In this study, the importance of origami in transforming 2D patterns into 3D objects is highlighted. Within a design workshop, projects of architecture graduate students were evaluated, focusing on the conversion of 2D patterns to 3D using various tools. The workshop consisted of two stages. In the first stage, students created basic geometric shapes such as spheres, cubes, and ellipses using the origami technique with A4 and A5 papers, and transformed them from 2D to 3D through practical tessellation processes. In the second stage, students were instructed in basic folding techniques using Rhino-Grasshopper software, enabling them to digitally model the transformation from 2D tessellation to 3D structures using the origami technique. The study examined these processes, considering variations in tessellation methods, the influence of material knowledge on tessellation processes, and the impact of material characteristics on form finding. [ABSTRACT FROM AUTHOR]

In Part I of this study, it was shown that all the three known types of monohedral hexagonal tessellations of the plane, those composed of equal irregular hexagons, have just a single deformation mode when tiles are considered as rigid bodies hinged to each other along the edges. A gallery of tessellated plates was simulated numerically to demonstrate the range of achievable deformed shapes. In Part II, the displacement field was first derived and a continuous interpolant for each type of tessellated plate. It turns out that all corresponding metasurfaces are described by quadrics. Afterwards, a parametric analysis was carried out to determine the effect of varying angles and edge lengths on the curvature, and the values of the geometric Poisson ratio of the plates. Finally, a method of fabrication is proposed based on the additive manufacturing of stiff tiles of negligible deformability and flexible connectors. Using this modular technique, it is possible to join together different monohedral tessellated plates able to deform into piece-wise quadrics. The nodal positions in the deformed configuration of the realized plates are measured after enforcing one principal curvature to assume a chosen value. The estimate of the other principal curvature confirms the analytical predictions. The presented tessellated plates permit to realize doubly curved shape-morphing metasurfaces with assorted shapes, which also can feature a certain surface roughness, and they can be employed in all applications demanding high surface accuracy and few actuators or just one. [ABSTRACT FROM AUTHOR]

A compact and tractable two-dimensional model to generate the topological network structure of domain walls in BiFeO3 thin films is presented in this study. Our method combines the stochastic geometry parametric model of the centroidal Voronoi tessellation optimized using the von Neumann entropy, a novel information-theoretic tool for networks. The former permits the generation of image-based stochastic artificial samples of domain wall networks, from which the network structure is subsequently extracted and converted to the graph-based representation. The von Neumann entropy, which reflects information diffusion across multiple spatiotemporal scales in heterogeneous networks, plays a central role in defining a fitness function. It allows the use of the network as a whole rather than using a subset of network descriptors to search for optimal model parameters. The optimization of the parameters is carried out by a genetic algorithm through the maximization of the fitness function and results in the desired graph-based network connectivity structure. Ground truth empirical networks are defined, and a dataset of network connectivity structures of domain walls in BiFeO3 thin films is undertaken through manual annotation. Both a versatile tool for manual network annotation of noisy images and a new automatic network extraction method for highquality images are developed. [ABSTRACT FROM AUTHOR]

The control of the shape of complex metasurfaces is a challenging task often addressed in the literature. This work presents a class of tessellated plates able to deform into surfaces of preprogrammed shape upon activation by any flexural load and that can be controlled by a single actuator. Quadric metasurfaces are obtained from infinitesimal origami maps of monohedral hexagonal tessellations of the plane, that is pavings in which all tiles are congruent to each other. Monohedral tessellated portions can be joined together to obtain more complex shapes, which can be locally synclastic or anticlastic and can have a certain roughness. We broaden previous work by providing a complete characterization of all the three known types of monohedral tessellations composed by irregular hexagons. The proposed two-dimensional structures may have applications in prosthetics, tissue engineering, wearable devices, energy harvesting devices, tunable focus mirrors and adaptive facades. The study is divided in two parts. In Part I, after introducing the discrete kinematics of tessellated plates, it is proved analytically that essentially each type of monohedral hexagonal tessellation possesses only one deformation mode. Afterwards, several numerical examples are provided to demonstrate the variety of achievable surface shapes. In Part II, first the metasurfaces corresponding to assigned tile geometries are given a continuum description, which establishes that the continuous interpolant is always a quadric. Then, experimental results on fabrication, assembly and surface accuracy are reported. [ABSTRACT FROM AUTHOR]

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E 2 , S 2 , and H 2 , by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H 2 by curved hyperbolic equilateral triangles whose vertex angles are 2 π / d for d = 7 , 8 , 9 , ... On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H 3 . We also show that a regular tessellation of H 3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H 4 . If n > 4 , then there exists no regular tessellation of H n . [ABSTRACT FROM AUTHOR]

Based on a newly proposed spatial data model Spatial Chromatic Model (SCM), we developed a spatial coding scheme, called the full-coded Ordinary Arranged Chromatic Diagram (full-OACD). As a type of spatial tessellation, full-OACD partitions a geographic space into a number of subspaces, such as cells, edges, and vertices. These subspaces are called spatial particles and are assigned with unique codes chromatic codes. The generation, structure, computation, and properties of full-OACD are introduced. Relations between particulate chromatic codes and spatial topology are investigated. Full-OACD is a kind of new discrete spatial coordinate system where the information of real-world entities is embedded. Full-OACD provides an informative and meaningful spatial coding framework for spatial topological analysis and many other potential applications in geospatial information science. [ABSTRACT FROM AUTHOR]

Generating space-filling arrangements of most discrete polyhedra nanostructures of the same shape is not possible. However, if the appropriate individual building blocks are selected (e.g., cubes), or multiple shapes of the appropriate dimensions are matched (e.g., octahedra and tetrahedra) and their pairing interactions are subsequently forced, space-filled architectures may be possible. With flexible molecular ligands (polyethylene glycol-modified DNA), the shape of a polyhedral nanoparticle can be deliberately altered and used to realize geometries that favor space tessellation. In this work, 10 new colloidal crystals were synthesized from DNA-modified nanocrystal building blocks that differed in shapes and sizes, designed to form space-filling architectures with micron-scale dimensions. The insights and capabilities provided by this new strategy substantially expand the scope of colloidal crystals possible and provide an expanded tool kit for researchers interested in designing metamaterials. [ABSTRACT FROM AUTHOR]

Curved displays can adjust their shape to accommodate different objects and are used in electronics and decorative lighting. Due to the immutable pixel spacing, existing commercial curved displays are flexible but not compatible with undevelopable surfaces. Inspired by kirigami and auxetic structures, we propose an approach that combines luminescent elements and rotating square tessellations to create a stretchable, arbitrary curve adaptive display. We connect square islands by vertical interconnects to relieve the stress concentration and provide extra deformation patterns. The vertical interconnects are patterned on a flexible printed circuit board (FPCB) using laser cutting and folded up via specially designed molds. Further, the freed-up space by folded interconnects allows the structure to be compressed. A prototype stretchable display is demonstrated that it can maintain electrical performance under biaxial strain and adapt to different Gaussian curvature surfaces, including cylindrical, spherical, saddle and arbitrary surfaces. Theoretical models and finite element calculations are established to describe the tensile behavior of the structures under different boundary conditions and agree with the experimental results. This proposed technology paves a feasible solution of mass production of adaptive curved displays and sets the trend for the next-generation display. [ABSTRACT FROM AUTHOR]

The age-old challenge of polygon quadrature involves converting a polygon into a square of equal area. In this educational resource, we utilize pattern blocks, commonly employed instructional aids in K-12 education across the United States, to visually demonstrate the transformation of different equilateral and regular pattern block polygons into squares. This is achieved through the application of the area conservation principle and geometric congruence/similarity reasoning. [ABSTRACT FROM AUTHOR]

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, n E . The effective electrical conductivity is proportional to n E when n E ≫ 1. [ABSTRACT FROM AUTHOR]

In this article we consider uniform tessellations of the hyperbolic plane. Two counts are performed. The first, considers the number of polygons in layers of the hyptagrid and is shown to be related to Fibonacci numbers. The second, considers the number of fishes in layers of a superimposed octagrid on Escher's circle limit III. To excute these counts we solve two linear recurrence relations, homogeneous and non-homogeneous. The initial conditions are set up by performing tessellations using a software in hyperbolic geometry. [ABSTRACT FROM AUTHOR]

Assessment of the seismic damage of coal mine overburden (OB) dump slopes gets compromised due to negligence of the consideration of heterogeneity in the shape and size of its particles. In this paper, in situ material parameters for an OB dump slope were evaluated through multichannel analysis of surface waves (MASW) investigations performed at the Jambad opencast coal mine of India. To realistically simulate the heterogeneity of the OB dump materials, Voronoi tessellation scheme coupled with extended finite-element method (XFEM) (using RS2 software) was employed. Behavior of a double-benched OB dump slope under earthquake excitations with a wide range of strong motion parameters (such as peak ground acceleration, predominant frequency, bracketed duration, and Arias intensity) was investigated through XFEM. The influence of strong motion parameters on the seismic behavior of a double-benched OB dump slope was reported in terms of transient (during the earthquake motion) and permanent deformation (after the earthquake motion) at various key points of the slope. Finally, the levels of damage experienced by the OB dump slope under earthquake ground motions were assessed qualitatively. The study would be advantageous for the mining industry in framing the safety guidelines for OB dump slopes in coal mine areas susceptible to earthquakes. The performance of a double-benched coal mine overburden (OB) dump slope under earthquake excitations with a wide range of strong ground motion parameters (such as peak ground acceleration, predominant frequency, bracketed duration, and Arias intensity) was investigated by employing Voronoi tessellation scheme coupled with extended finite-element method (XFEM). The influence of strong ground motion parameters on the seismic behavior of the double-benched OB dump slope was reported in terms of transient (during the earthquake motion) and permanent or residual deformation of the slope at various critical points. Further, the levels of damage experienced by the OB dump slope under earthquake ground motions were assessed qualitatively. This research work may provide guidance to practicing engineers for considering the heterogeneity of the OB dump slope while assessing its seismic performance. The study is going to be useful for the design of OB dump slopes for coal mines located in a seismically active region. If the site-specific earthquake parameters are available, the study may provide important aspects of seismic design of the OB dump slope. [ABSTRACT FROM AUTHOR]

This study investigates sixth-grade students' explorations of tessellations with Scratch through the mathematical aesthetics. For this purpose, the study analyzed the students' explorations by considering the roles of mathematical aesthetics identified by Sinclair (2004): generative, evaluative, and motivational role. Six middle-school students were asked to create regular and semi-regular tessellations by using Scratch and interviewed individually about the investigation of tessellations. The findings indicated that the students' generative aesthetic responses to mathematical tasks were most evident in their explorations with Scratch. The students recognized the aesthetic value of mathematical entities while working on tessellations. They never gave up studying on tessellations with Scratch and did not need any external motivation to complete the tasks. Implications for research and practice for teaching are drawn. [ABSTRACT FROM AUTHOR]

The geometry-architecture relationship, which plays a limiting role in the building construction process, is an important concept in determining the identity of the buildings. Tessellations, which differentiate the structures from each other in the architectural design process, are among the methods that can be considered within the geometry-architecture relationship. In the world of modern architecture, it is possible to come across tessellation configurations in the formation of buildings, their facades, and load-bearing system designs. In this study, the reflections of tessellation patterns on steel load-bearing system designs were examined. In the first stage, the load-bearing patterns were analyzed by performing a situation analysis through the real structures with different tessellation configurations. Eight tessellation patterns were created with the most used geometries, and their geometric characteristics, structural performances, and costs were investigated. The models designed with the Grasshopper software were evaluated with the Sap2000 analysis program. To create a comparable structural framework and to link the tessellation patterns with the steel load-bearing systems, planar and single-curvature vault grid systems, in which symmetric/periodic types of tessellation can be applied, were used. Depending on the variation in tessellation modulations, optimum solutions were sought in load-bearing structure systems arranged according to grid density and grid factor values. As a result of structural analyses, systems that offer optimum solutions were shown with comparative analyses. Then the performances of geometric designs were discussed. Considering the cost-based scoring of the systems because of optimization, triangle-rectangle (3.3.3.4.4) in planar structures, rectangle (4.4.4.4) and triangle–hexagon (3.3.3.3.6) tessellation systems in single vault structures come to the fore. [ABSTRACT FROM AUTHOR]

Origami tessellations, whose crease pattern has translational symmetries, have attracted significant attention in designing the mechanical properties of objects. Previous origami-based engineering applications have been designed based on the "uniform-folding" of origami tessellations, where the folding of each unit cell is identical. Although "nonuniform-folding" allows for nonlinear phenomena that are impossible through uniform-folding, there is no universal model for nonuniform-folding, and the underlying mathematics for some observed phenomena remains unclear. Wavy folded states that can be achieved through nonuniform-folding of the tubular origami tessellation called a waterbomb tube are an example. Recently, the authors formulated the kinematic coupled motion of unit cells within a waterbomb tube as the discrete dynamical system and identified a correspondence between its quasiperiodic solutions and wavy folded states. Here, we show that the wavy folded state is a universal phenomenon that can occur in the family of rotationally symmetric tubular origami tessellations. We represent their dynamical system as the composition of the two 2D mappings: taking the intersection of three spheres and crease pattern transformation. We show the universality of the wavy folded state through numerical calculations of phase diagrams and a geometric proof of the system's conservativeness. Additionally, we present a non-conservative tubular origami tessellation, whose crease pattern includes scaling. The result demonstrates the potential of the dynamical system model as a universal model for nonuniform-folding or a tool for designing metamaterials. [ABSTRACT FROM AUTHOR]

Traditional tomographic methods do not consider the uncertainties associated with near-surface velocities and static corrections and provide a deterministic solution to the estimation problem. However, these uncertainties significantly affect structural mapping and interpretation of seismic imaging results. On the other hand, Bayesian first-arrival tomography provides multiple near-surface models that fit observed traveltimes equally well and enable the study of potential solution distributions. We demonstrate this approach on a complex synthetic near-surface model, representative of arid environments, to quantify associated velocity and statics uncertainties. We evaluate two different parameterizations for subsurface velocities in the context of near-surface Bayesian tomography: Voronoi tessellation with natural neighbor interpolation and the more conventional Delaunay triangulation with linear interpolation. Our analysis shows that the Voronoi cell parameterization with natural neighbor interpolation is more appropriate for this problem. Finally, the new approach is applied to compare two alternative acquisition geometries comprising conventional surface receivers and surface receivers augmented with vertical receiver arrays. The results demonstrate that adding vertical receiver arrays to conventional surface receivers can significantly reduce the near-surface velocity uncertainty and thus increases the accuracy of the seismic imaging results. Furthermore, the study shows that Bayesian tomography can be used as a tool for evaluating different source and receiver geometries during the acquisition design stage. [ABSTRACT FROM AUTHOR]

Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. These techniques have wide applications in fields such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) k -connected graphs with the property that for any j vertices there is a longest path avoiding all of them? This question of Zamfirescu (1972) was the first variant of Gallai's question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfirescu's question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufficiently smaller orders. The problem of finding nonconcurrent longest paths in Archimedean tessellations refers to finding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. The complexity of embedding graph is still unknown. This problem can be challenging because it requires finding paths that are both long and do not intersect, which can be difficult due to the constraints of the lattice structure. [ABSTRACT FROM AUTHOR]

Tessellation is a mathematical concept used in geometric design. As in many fields, this method is also encountered in architecture. In this study, it is aimed to investigate the tessellation configurations in marble and wooden-columned mosques within the scope of geometric design in architecture, to show the effectiveness of geometry in the architectural production in these buildings, and to draw attention to the relationship between tessellation and geometry in Islamic buildings. The research question of the study is how tessellation configurations are applied in marble and wooden-columned mosques in Islamic architecture and how this relationship is established. Depending on this research question, Beyşehir Eşrefoğlu Mosque, Sivrihisar Ulu Mosque, and Konya Alâeddin Mosque in Turkey were examined within the scope of this study. The common point of these Islamic buildings is the use of tessellation patterns in the minbar and mihrab sections. Content analysis was performed by looking at the architectural features of the marble and wooden-columned mosques and geometric analysis of the tessellation configurations in the minbars and mihrabs was conducted with the data obtained. As a result, it has been seen that various variations in tessellation configurations are used in marble and wooden-columned mosques, which are products of Islamic architecture. The academic study in which the tessellation method is emphasized in mosques is limited. Therefore, this study will fill the gap in the literature. [ABSTRACT FROM AUTHOR]

Copyright of Rozwój Regionalny & Polityka Regionalna is the property of Uniwersytetu im. Adama Mickiewicza (IH UAM) 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.)

Copyright of VLC Arquitectura is the property of VLC Arquitectura 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.)

In the planning stage of the fabrication process of physical models of cellular structures, a surface model of the structure needs to be adjusted to acquire the requisite properties, but errors emerge frequently at this stage. The main objective of this research was to repair or reduce the impact of deficiencies and errors before the fabrication of physical models. For this purpose, it was necessary to design models of cellular structures with different accuracy settings in PTC Creo and then compare them after the tessellation process using GOM Inspect. Subsequently, it was necessary to locate the errors occurring in the process of preparing models of cellular structures and propose an appropriate method of their repair. It was found that the Medium Accuracy setting is adequate for the fabrication of physical models of cellular structures. Subsequently, it was found that within regions where mesh models merged, duplicate surfaces emerged, and the entire model could be considered as manifesting non-manifold geometry. The manufacturability check showed that in the regions with duplicate surfaces inside the model, the toolpath creation strategy changed, causing local anisotropy within 40% of the fabricated model. A non-manifold mesh was repaired in the proposed manner of correction. A method of smoothing the model's surface was proposed, reducing the polygon mesh density and the file size. The findings and proposed methods of designing cellular models, error repair and smoothing methods of the models can be used to fabricate higher-quality physical models of cellular structures. [ABSTRACT FROM AUTHOR]

An animal is a planar shape formed by attaching congruent regular polygons along their edges. Usually, these polygons are a finite subset of tiles of a regular planar tessellation. These tessellations can be parameterized using the Schläfli symbol { p , q } , where p denotes the number of sides of the regular polygon forming the tessellation and q is the number of edges or tiles meeting at each vertex. If (p - 2) (q - 2) > 4 , = 4 , or < 4 , then the tessellation corresponds to the geometry of the hyperbolic plane, the Euclidean plane, or the sphere, respectively. In 1976, Harary and Harborth studied animals defined on regular tessellations of the Euclidean plane, finding extremal values for their vertices, edges, and tiles, when any one of these parameters is fixed. They named animals attaining these extremal values as extremal animals. Here, we study hyperbolic extremal animals. For each { p , q } corresponding to a hyperbolic tessellation, we exhibit a sequence of spiral animals and prove that they attain the minimum numbers of edges and vertices within the class of animals with n tiles. We also give the first results on enumeration of extremal hyperbolic animals by finding special sequences of extremal animals that are unique extremal animals, in the sense that any animal with the same number of tiles which is distinct up to isometries cannot be extremal. [ABSTRACT FROM AUTHOR]

• Influence of clockwise and anti-clockwise chiralisation on the mechanical properties of axially asymmetric Euclidean tessellations was investigated. • Novel chiral metamaterials based on pentagonal and hexagonal tessellations were produced. • Results show that the Poisson's ratio and Young's modulus is highly dependent on the direction of chiralisation. Chiral honeycombs are a class of auxetic metamaterials which are characterised by a lack of plane symmetry. Besides the traditional chiral honeycombs based on regular monohedral tessellations such as the hexachiral and tetrachiral honeycombs, a vast range of auxetic chiral metamaterials may be produced through the 'chiralisation' of Euclidean tessellations made from polyhedral tilings and/or irregular monohedral polygons. In this work, we show for the first time, how the direction of the geometric chiral transformation, i.e. clockwise vs anti-clockwise chiralisation, has an influence on the mechanical properties and deformation modes of the resultant chiral metamaterial systems. This influence, which is a result of the intrinsic absence of axial symmetry in the original base tessellation, can lead to the production of two unique and distinct chiral metamaterial configurations with completely different mechanical properties originating from a single base tessellation. In this work we demonstrate and quantify, through a wide range of numerical simulations and experimental tests on additively-manufactured prototypes, the effect of the direction of chiralisation on two tessellations: a Florent pentagonal system with hexagonal rotational symmetry and a hexagonal monohedral tessellation with trigonal rotational symmetry. The results obtained show that the clockwise and anti-clockwise chiral structures exhibit significantly different mechanical properties and deformation modes, highlighting the increased versatility of this relatively novel class of chiral metamaterials. [Display omitted] [ABSTRACT FROM AUTHOR]

Self-assembly of trans-{EuI2} nodes and ditopic ligands leads to isoreticular 2D frameworks featuring a rare, non-kagome Archimedean tessellation. The topology and intra-layer Eu(II)–Eu(II) antiferromagnetic interactions provide the prerequisites for geometrical spin frustration, which, due to the spin state degeneracy, is key for novel phenomena such as enhanced magnetic refrigeration. [ABSTRACT FROM AUTHOR]

Engineered granular materials have gained considerable interest in recent years. For this substance, the primary design variable is grain shape. Optimizing grain form to achieve a macroscopic property is difficult due to the infinite-dimensional function space particle shape inhabits. Nonetheless, by parameterizing morphology the dimension of the problem can be reduced. In this work, we study the effects of both intuitive and machine-picked shape descriptors on granular material properties. First, we investigate the effect of classical shape descriptors (roundness, convexity, and aspect ratio) on packing fraction ϕ and coordination number Z. We use a genetic algorithm to generate a uniform sampling of shapes across these three shape parameters. The shapes are then simulated in the level set discrete element method. We discover that both ϕ and Z decrease with decreasing convexity, and Z increases with decreasing aspect ratio across the large sampling of morphologies—including among highly non-convex grains not commonly found in nature. Further, we find that subtle changes in mesoscopic properties can be attributed to a continuum of geometric phenomena, including tessellation, hexagonal packing, nematic order and arching. Nonetheless, such descriptors alone can not entirely describe a shape. Thus, we find a set of 20 descriptors which uniquely define a morphology via deep generative models. We show how two of these machine-derived parameters affect ϕ and Z. This methodology can be leveraged for topology optimization of granular materials, with applications ranging from robotic grippers to materials with tunable mechanical properties. [ABSTRACT FROM AUTHOR]

The objective of this study is to develop generalized empirical closed‐form equations to predict the compressive mechanical properties and determine geometrical parameters. To achieve that, 117 models are built and analyzed using ABAQUS/CAE 2016 to provide two types of reliable data: one for lattice mechanical properties based on finite element method and the other for geometrical parameters using the measurements of ABAQUS diagnostic tool. All the models are created by modifying the basic feature of body‐centered cubic lattice structure based on a range of strut angles, a set of relative densities, and two design sets. Also, the influence of lattice cell tessellations and material distribution at strut intersections are considered within these models to provide accurate results. The first data set is fitted with the scaling laws, relating relative elastic modulus and stress with the relative density, to determine Gibson and Ashby's coefficients. The second type of data regarding lattice geometries is correlated with the relative density to estimate actual lattice volume, strut radius, aspect ratio, and overall lattice volume. By this way, these equations can be used to predict directly the lattice characteristics and geometrical parameters without the need for ABAQUS. The results show that the generalized empirical closed‐form equations can predict well both the lattice characteristics and geometries. In addition, the relative stresses and elastic modulus increase with increasing the strut angles since the main deformation mechanisms move toward stretch‐dominated rather than bending. Besides, Gibson and Ashby's coefficients along with the geometrical factors of aspect ratios are found to be approximately similar for both generations. This study contributes to developing efficient equations to provide the researchers with a preliminary insight about the best lattice design and its compatibility in a certain application before starting the fabrication process. [ABSTRACT FROM AUTHOR]

Given a set R of robots, each one located at a different vertex of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that | R | = | F | , APF asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections. So far, as possible graph discretizing the Euclidean plane only the standard square grid has been considered in the context of the classical OBLOT model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. In particular, the former can be considered as the most general in terms of possible symmetries and trajectories. We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph. The algorithm and its correctness are based on a rigorous methodology. • We solve the Arbitrary Pattern Formation problem for asynchronous robots moving on triangular grids. • We show that the provided algorithm works also for square and hexagonal grids. • The algorithm is able to create patterns with possible multiplicities. • The algorithm and its correctness are based on a rigorous methodology. [ABSTRACT FROM AUTHOR]

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions. [ABSTRACT FROM AUTHOR]

Let \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [ABSTRACT FROM AUTHOR]

Contour detection has a rich history in multiple fields such as geography, engineering, and earth science. The predominant approach is based on piecewise planar tessellation and now being challenged concerning the extraction of contour objects for non-linear elevation functions, particularly with respect to bicubic spline functions. A storage-efficient method was developed in previous research, but the detection of the complete set of contour objects is yet to be realized. Although intractable, theoretical underpinnings pertinent to curvature resulted in an approach to realize the complete detection of objects. Given a digital elevation model dataset, in this study, a bicubic spline surface function was first determined. Thereafter, candidate initial points on the edges across the region of interest were identified, and the recursive disaggregation of rectangles was repeated if the non-existence of a solution could not be assured. A developed tracking method was then applied. During advancement, other initial points on the same contour curve were identified and eliminated to circumvent duplicate detection. The completeness of the outlets provides analytical tools for elevation and other geographical assessments. Demonstrative experiments included the development of a three-dimensional contour-based network and slope assessments. The latter application transforms the slope analysis type from raster-based to vector-based. Detection of a complete set of contour objects amenable to bicubic spline surfaces. Small closure inside a single patch is detectable if size exceeds the standard. Curvature & tolerances central to step length adjustment and tangent angle determination. Redundant initial points are identified and eliminated during the tracking process. Various potential applications in addition to geographical elevations. [ABSTRACT FROM AUTHOR]

Many 16th-century Castilian coffered ceilings combine a long tradition of Hispanic carpentry with the introduction of polygonal coffers, typical of the Renaissance period. In addition, these structures are made up of several panels. This polyhedral shape creates an impression of greater depth and enhances the elegance of the space. However, it considerably complicates its construction. It must be borne in mind that in the rest of Europe these Renaissance structures are horizontal. The combination of flat polygons and multi-sided ceilings requires imaginative geometric solutions to render its corners. The ceiling of the main hall of the Monastery of Uclés is a Renaissance coffered ceiling made up of octagons and squares. This pattern covers both the flat and the pitched panels. In this case, the designer has inserted hexagons that connect adjacent octagons to render each of the corners. This research studies the context and analyzes the definition of this unique solution. [ABSTRACT FROM AUTHOR]

Uniaxial compression tests on Hydro-StoneTB® porous cubes were numerically modeled using a distinct element program (UDEC) for two-dimensional geological material simulation. Under uniaxial compression, numerical square models with varying void porosity and geometry, identical to an experimental testing, were simulated. The open-ended longitudinal openings in the experimental specimens, cubic porous Hydro-StoneTB®, were of various forms, sizes, uniformity, and distributions. UDEC's Voronoi tessellation joint generator was used to simulate the material of the experimental specimens as an assembly of discrete units interacting along their boundaries. The results showed that the numerical results from the two-dimensional numerical analysis utilizing the discrete element method, UDEC program, were consistent with the experimental data. However, when compared to the experimental values, the numerical analysis produced conservative values for both uniaxial compressive strength and Young's modulus due to the modeling of a three-dimensional material in a two-dimensional planar strain. Furthermore, regardless of void porosity or geometry, the tension (axial splitting) failure mode is the dominating failure mode of rock-like materials including voids, similar to the experimental results. It was also demonstrated that using a power relationship, two-dimensional numerical data can be converted to three-dimensional experimental results. [ABSTRACT FROM AUTHOR]

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V (G) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E (G). A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t (G) is the size of a minimum total tessellation cover of G. We present lower bounds T t (G) ≥ ω (G) and T t (G) ≥ s (G) + 1 , where ω (G) is the size of a maximum clique, and s (G) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t (G) = ω (G) or T t (G) = s (G) + 1. We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t (G) ≤ k. We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, (2 , 1) -chordal graphs and planar graphs. [ABSTRACT FROM AUTHOR]

In contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set B , an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in B and, in fact, all hyperplanes intersecting B. Therefore the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are also hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties. [ABSTRACT FROM AUTHOR]

Irregular 3D biological scaffolds have been widely observed in nature. Therefore, in the current work, new designs are proposed for lightweight 3D scaffolds based on Voronoi tessellation with high porosity. The proposed designs are inspired by nature, which has undoubtedly proven to be the best designer. Thus, the Rhinoceros 7/Grasshopper software was used to design three geometric models for both normal and elongated Voronoi structures: homogeneous, gradient I, and gradient II. Then, stereolithography (SLA) additive manufacturing was utilized to fabricate biopolymeric materials. Finally, a compression test was carried out to study and compare the mechanical properties of the designed samples. The gradient I cylinder show the highest Young's modulus. For the Homogeneous and gradient II cylinders, elongated Voronoi structures show superior mechanical properties and energy absorption compared to normal Voronoi designs. Hence, these designs are promising topologies for future applications. [ABSTRACT FROM AUTHOR]