Planar Tessellations Research Articles

Modeling Crack Patterns by Modified STIT Tessellations

Random planar tessellations are presented which are generated by subsequent division of their polygonal cells. The purpose is to develop parametric models for crack patterns appearing at length scales which can change by orders of magnitude in areas such as nanotechnology, materials science, soft matter, and geology. Using the STIT tessellation as a reference model and comparing with phenomena in real crack patterns, three modifications of STIT are suggested. For all these models a simulation tool, which also yields several statistics for the tessellation cells, is provided on the web. The software is freely available via a link given in the bibliography of this article. The present paper contains results of a simulation study indicating some essential features of the models. Finally, an example of a real fracture pattern is considered which is obtained using the deposition of a thin metallic film onto an elastomer material – the results of this are compared to the predictions of the model.

Exponential Moments for Planar Tessellations

In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

Stress Concentration in Low-Porosity Periodic Tessellations With Generic Patterns of Elliptical Holes Under Biaxial Strain

Stress concentration in porous materials is one of the most crucial culprits of mechanical failure. This paper focuses on planar porous materials with porosity less than 5%. We present a stress-prediction model of an arbitrarily rotated elliptical hole in a rhombus shaped representative volume element (RVE) that can represent a class of generic planar tessellations, including rectangular, triangular, hexagonal, Kagome, and other patterns. The theoretical model allows the determination of peak stress and distribution of stress generated near the edge of elliptical holes for any arbitrary tiling under displacement loading and periodic boundary conditions. The results show that the alignment of the void with the principal directions minimizes stress concentration. Numerical simulations support the theoretical findings and suggest the observations remain valid for porosity as large as 5%. This work provides a fundamental understanding of stress concentration in low-porosity planar materials with insight that not only complements classical theories on the subject but also provides a practical reference for material design in mechanical, aerospace, and other industry.

Planar tessellations that have the half-Gilbert structure

Abstract In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

An overview of curvature bounds and spectral theory of planar tessellations

We give a survey about the spectral consequences of upper bounds on the curvature on planar tessellating graphs. We first discuss spectral bounds and then put a particular focus on uniformly decreasing curvature. This case is characterized by purely discrete spectrum and we further present eigenvalue asymptotics and exponential decay of eigenfunctions. We then discuss absence of compactly supported eigenfunctions and dependence of the spectrum of the Laplacian on the underlying ` space.

Visualizing Research of Construction of Planar Dynamic Systems with Truncated Fourier Series

The paper constructs the planar dynamic systems with truncated Fourier series. We propose a method to generate planar tiling images of the iterative mapping with truncated Fourier series. It constructed the planar tessellations of planar dynamic systems with cyclic windows. The iterative mapping is constructed. Then we generate cyclic windows of the dynamic plane with these mappings. Filled-in Julia sets are created by generating the coordinates of any cyclic windows. The images in different windows are continuous. We can choose any cyclic windows as the basic computing region and stretch it into a square, which is transferred to the plane to compose the new form chaos attractor and filled-in Julia sets images in the Euclidian plane. Experimental results show the effectiveness of the proposed approach.

On the [formula omitted] spectrum of Laplacians on graphs

We study the p-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion without any further assumptions. We study applications to normalized Laplacians including symmetries of the spectrum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians on planar tessellations for which we relate the spectral p-independence to assumptions on the curvature.

On Planar Tessellations and Interference Estimation in Wireless Ad-Hoc Networks

In wireless networks, the received signal power is related to the distance between the transmitter and receiver. Planar tessellations, such as square and hexagonal tessellations, have found a wide range of applications in the analysis of wireless networks. A geometric probabilistic model is considered for interference estimation in wireless ad-hoc networks, utilizing the distance distributions for hexagonal and square tessellations. The impact of employing square and hexagonal regions is analyzed. Simulation results demonstrate that both models are accurate.

Construction of Planar Dynamic Systems with Function of Non-Linear Angle Variables

This paper constructs the planar dynamic systems with non-equidistant cyclic windows. To generate planar tiling images with different visual effects by the same iterative mapping, we propose a method that allows the planar tessellations to be constructed by a planar dynamic system with non-equidistant cyclic windows. A family of iterative mappings are constructed which generate cyclic windows of variant size in the dynamic plane by incorporating cosine functions and non-linear angle variables with the same parameters. Filled-in Julia sets in different cyclic windows are created by generating the coordinates of any cyclic windows. The images in different windows are continuous but with individual structures. We can choose any cyclic windows as the basic computing region and stretch or compress it into a square, which is transferred to the plane to compose the planar tilling. Experimental results show the effectiveness of the proposed approach.

Topological relationships in spatial tessellations

Tessellations of R3 that use convex polyhedral cells to fill the space can be extremely complicated. This is especially so for tessellations which are not ‘facet-to-facet’, that is, for those where the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. Adjacency concepts between neighbouring cells (or between neighbouring cell elements) are not easily formulated when facets do not coincide. In this paper we make the first systematic study of these topological relationships when a tessellation of R3 is not facet-to-facet. The results derived can also be applied to the simpler facet-to-facet case. Our study deals with both random tessellations and deterministic ‘tilings’. Some new theory for planar tessellations is also given.

Topological relationships in spatial tessellations

Tessellations of R 3 that use convex polyhedral cells to fill the space can be extremely complicated. This is especially so for tessellations which are not ‘facet-to-facet’, that is, for those where the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. Adjacency concepts between neighbouring cells (or between neighbouring cell elements) are not easily formulated when facets do not coincide. In this paper we make the first systematic study of these topological relationships when a tessellation of R 3 is not facet-to-facet. The results derived can also be applied to the simpler facet-to-facet case. Our study deals with both random tessellations and deterministic ‘tilings’. Some new theory for planar tessellations is also given.

MOMENTS OF THE LENGTH OF LINE SEGMENTS IN HOMOGENEOUS PLANAR STIT TESSELLATIONS

Homogeneous planar tessellations stable under iteration (STIT tessellations) are considered. Using recent results about the joint distribution of direction and length of the typical I-, K- and J-segment we prove closed formulas for the first, second and higher moments of the length of these segments given their direction. This especially leads to themean values and variances of these quantities andmean value relations as well as general moment relationships. Moreover, the relation between these mean values and certain conditional mean values (and also higher moments) is discussed. The results are also illustrated for several examples.

Finding Auxetic Frameworks in Periodic Tessellations

It appears that most models for micro-structured materials with auxetic deformations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent approach of finding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms, which have Poisson's ratio νss=-1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices. One of these, baptized Triangle-Square Wheels, has been produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective Poisson's ratio νLE≈-0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios νSS of the skeletal structure and νLE of the linear-elastic cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.

Inhomogeneous random planar tessellations generated by lines

Random planar tessellations in bounded convex windows are generated by dividing random cells with random lines. It is suggested that the random STIT tessellations of Nagel and Weiss, if restricted to a bounded convex window, can be interpreted as a special case.

Physarum Boats: If Plasmodium Sailed It Would Never Leave a Port

Plasmodium ofPhysarum polycephalumis a single huge (visible by naked eye) cell with a myriad of nuclei. The plasmodium is a promising substrate for non-classical, nature-inspired computing devices. It is capable of approximation of the shortest path in a maze, computation of planar proximity graphs and plane tessellations, primitive memory and decision making. The unique properties of the plasmodium make it an ideal candidate for a role of amorphous biological robots with massive parallel information processing and distributed inputs and outputs. We show that when adhered to a lightweight object resting on a water surface the plasmodium can propel the object by oscillating its protoplasmic pseudopodia. In experimental laboratory conditions and computational experiments we study phenomenology of the plasmodium-floater system, and possible mechanisms of controlling motion of objects propelled by on-board plasmodium.

The essential spectrum of the Laplacian on rapidly branching tessellations

In this paper, we characterize absence of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover, we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature.

Joint distribution of direction and length of typical I-segment in a homogeneous random planar tessellation stable under iteration

The paper deals with homogeneous random planar tessellations stable under iteration (random STIT tessellations). The length distribution of the typical I-segment is already known in the isotropic case [8]. In the present paper, the anisotropic case is treated. Then also the direction of the typical I-segment is of interest. The joint distribution of direction and length of the typical I-segment is evaluated. As a first step, the corresponding joint distribution for the so-called typical remaining I-segment is derived.

A global construction of homogeneous random planar tessellations that are stable under iteration

Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation in the whole plane. So far, an explicit geometrical construction for those tessellations was only known within bounded windows.

Length distributions of edges in planar stationary and isotropic STIT tessellations

Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given.

Multisided arrays of control points for multisided Bézier patches

A survey is presented examining a variety of different techniques, some old and some new, for extending the construction of standard three sided and four sided Bézier patches to n-sided surface patches. Standard triangular and rectangular Bézier patches can be defined either explicitly using Bernstein blending functions or recursively using de Casteljau pyramid algorithms based on barycentric coordinate functions. Underpinning both of these constructions are control points organized into triangular and rectangular arrays. Thus to extend the notion of Bézier patches to multisided schemes, we need to construct multisided arrays of control points and we need to generalize either the Bernstein basis functions or the barycentric coordinate functions to polygonal domains. But what exactly are multisided arrays of control points, and how precisely do we construct Bernstein basis functions or barycentric coordinates for polygonal domains? There is no single answer to any of these questions: different answers lead to different types of multisided Bézier schemes. Here we focus on different techniques for indexing multisided arrays, including Greek gnomons, spider webs, planar tessellations, fractal gaskets, Minkowski sums, and lattice polygons. For each of these indexing sets, we provide, whenever possible, either the associated bivariate Bernstein blending functions or the barycentric coordinate functions and the associated de Casteljau pyramid algorithm for the corresponding polygonal domains.