Tessellation Of Space Research Articles

Characterizing the Local Primary Dendrite Arm Spacing in Directionally Solidified Dendritic Microstructures

Characterizing the spacing of primary dendrite arms in directionally-solidified microstructures is an important step for developing process-structure-property relationships by enabling the quantification of (i) the influence of processing on microstructure and (ii) the influence of microstructure on properties. In this work, we utilized a new Voronoi-based approach for spatial point pattern analysis that was applied to an experimental dendritic microstructure. This technique utilizes a Voronoi tessellation of space surrounding the dendrite cores to determine nearest neighbors and the local primary dendrite arm spacing. In addition, we compared this technique to a recent distance-based technique and a modification to this using Voronoi tesselations. Moreover, a convex hull-based technique was used to include edge effects for such techniques, which can be important for thin specimens. These methods were used to quantify the distribution of local primary dendrite arm spacings, their spatial distribution, and their correlation with interdendritic eutectic particles for an experimental directionally-solidified Ni-based superalloy micrograph. This can be an important step for correlating with both processing and properties in directionally-solidified dendritic microstructures.

Studying the elastic properties of nanocrystalline copper using a model of randomly packed uniform grains

We develop a new Voronoi protocol, which is a space tessellation method, to generate a fully dense (containing no voids) model of nanocrystalline copper with precise grain size control; we also perform uniaxial tensile tests using molecular dynamical (MD) simulations to measure the elastic moduli of the grain boundary and the grain interior components at 300K. We find that the grain boundary deforms more locally compared with the grain core region under thermal vibrations and is elastically less stiff than the core component at finite temperature. The elastic modulus of the grain boundary is lower than 30% of that of the grain interior. Our results will aid in the development of more accurate continuum models of nanocrystalline metals.

On subgroups of crystallographic Coxeter groups

A framework is presented based on color symmetry theory that will facilitate the determination of the subgroup structure of a crystallographic Coxeter group. It is shown that the method may be extended to characterize torsion-free subgroups. The approach is to treat these groups as groups of symmetries of tessellations in space by fundamental polyhedra.

Fractional Dynamics of Genetic Algorithms Using Hexagonal Space Tessellation

The paper formulates a genetic algorithm that evolves two types of objects in a plane. The fitness function promotes a relationship between the objects that is optimal when some kind of interface between them occurs. Furthermore, the algorithm adopts an hexagonal tessellation of the two-dimensional space for promoting an efficient method of the neighbour modelling. The genetic algorithm produces special patterns with resemblances to those revealed in percolation phenomena or in the symbiosis found in lichens. Besides the analysis of the spacial layout, a modelling of the time evolution is performed by adopting a distance measure and the modelling in the Fourier domain in the perspective of fractional calculus. The results reveal a consistent, and easy to interpret, set of model parameters for distinct operating conditions.

Families of tessellations of space by elementary polyhedra via retessellations of face-centered-cubic and related tilings

The problem of tiling or tessellating (i.e., completely filling) three-dimensional Euclidean space R(3) with polyhedra has fascinated people for centuries and continues to intrigue mathematicians and scientists today. Tilings are of fundamental importance to the understanding of the underlying structures for a wide range of systems in the biological, chemical, and physical sciences. In this paper, we enumerate and investigate the most comprehensive set of tilings of R(3) by any two regular polyhedra known to date. We find that among all of the Platonic solids, only the tetrahedron and octahedron can be combined to tile R(3). For tilings composed of only congruent tetrahedra and congruent octahedra, there seem to be only four distinct ratios between the sides of the two polyhedra. These four canonical periodic tilings are, respectively, associated with certain packings of tetrahedra (octahedra) in which the holes are octahedra (tetrahedra). Moreover, we derive two families of an uncountably infinite number of periodic tilings of tetrahedra and octahedra that continuously connect the aforementioned four canonical tilings with one another, containing the previously reported Conway-Jiao-Torquato family of tilings [Conway et al., Proc. Natl. Acad. Sci. USA 108, 11009 (2011)] as a special case. For tilings containing infinite planar surfaces, nonperiodic arrangements can be easily generated by arbitrary relative sliding along these surfaces. We also find that there are three distinct canonical periodic tilings of R(3) by congruent regular tetrahedra and congruent regular truncated tetrahedra associated with certain packings of tetrahedra (truncated tetrahedra) in which the holes are truncated tetrahedra (tetrahedra). Remarkably, we discover that most of the aforementioned periodic tilings can be obtained by "retessellating" the well-known tiling associated with the face-centered-cubic lattice, i.e., by combining the associated fundamental tiles (regular tetrahedra and octahedra) to form larger polyhedra.

Complete Topology of Cells, Grains, and Bubbles in Three-Dimensional Microstructures

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.

Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties

Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists and mathematicians for centuries. Here, we analytically construct the densest known packing of truncated tetrahedra with a remarkably high packing fraction φ = 207/208 = 0.995192..., which is amazingly close to unity and strongly implies its optimality. This construction is based on a generalized organizing principle for polyhedra lacking central symmetry that we introduce here. The "holes" in the putative optimal packing are perfect tetrahedra, which leads to a new tessellation of space by truncated tetrahedra and tetrahedra. Its packing characteristics and equilibrium melting properties as the system undergoes decompression are discussed.

Linking Microscopic Spatial Patterns of Tissue Destruction in Emphysema to Macroscopic Decline in Stiffness Using a 3D Computational Model

Pulmonary emphysema is a connective tissue disease characterized by the progressive destruction of alveolar walls leading to airspace enlargement and decreased elastic recoil of the lung. However, the relationship between microscopic tissue structure and decline in stiffness of the lung is not well understood. In this study, we developed a 3D computational model of lung tissue in which a pre-strained cuboidal block of tissue was represented by a tessellation of space filling polyhedra, with each polyhedral unit-cell representing an alveolus. Destruction of alveolar walls was mimicked by eliminating faces that separate two polyhedral either randomly or in a spatially correlated manner, in which the highest force bearing walls were removed at each step. Simulations were carried out to establish a link between the geometries that emerged and the rate of decline in bulk modulus of the tissue block. The spatially correlated process set up by the force-based destruction lead to a significantly faster rate of decline in bulk modulus accompanied by highly heterogeneous structures than the random destruction pattern. Using the Karhunen-Loève transformation, an estimator of the change in bulk modulus from the first four moments of airspace cell volumes was setup. Simulations were then obtained for tissue destruction with different idealized alveolar geometry, levels of pre-strain, linear and nonlinear elasticity assumptions for alveolar walls and also mixed destruction patterns where both random and force-based destruction occurs simultaneously. In all these cases, the change in bulk modulus from cell volumes was accurately estimated. We conclude that microscopic structural changes in emphysema and the associated decline in tissue stiffness are linked by the spatial pattern of the destruction process.

Micro-mechanical analysis of deformation characteristics of three-dimensional granular materials

The deformation characteristics of idealized granular materials have been studied from the micro-mechanical viewpoint, using Bagi’s three-dimensional micro-mechanical formulation for the strain tensor [Bagi, K., 1996. Mechanics of Materials 22, 165–177]. This formulation is based on the Delaunay tessellation of space into tetrahedra. The set of edges of the tetrahedra can be divided into physical contacts and virtual contacts between particles. Bagi’s formulation expresses the continuum, macro-scale strain as an average over all edges, of their relative displacements (between two successive states) and the complementary-area vectors. This latter vector is a geometrical quantity determined from the set of edges, i.e. from the structure of the particle packing. Results from Discrete Element Method simulations of isotropic and triaxial loading of a three-dimensional polydisperse packing of spheres have been used to investigate statistics of the branch vectors and complementary-area vectors of edges (subdivided into physical and virtual contacts) and of the relative displacements of edges. The investigated statistics are probability density functions and averages over groups of edges with the same orientation. It is shown that these averages can be represented by second-order Fourier series in edge orientation. Edge orientations are distributed isotropically, contrary to contact orientations. The average lengths of the branch vectors and the normal component of the complementary-area vectors are distributed isotropically (with respect to the edge orientation) and their average values are related to each other and to the volume fraction of the assembly. The other two components of the complementary-area vector are zero on average. The total deformation of the assembly, as given by the average of the relative displacements of the edges of the Delaunay tessellation follows the uniform-strain prediction. However, neither the deformation of the physical contact network nor of the virtual contact network has this property. The average relative displacement of physical edges in the normal direction (determined by the branch vector) is smaller than that according to the uniform-strain assumption, while that of virtual contacts is larger. This is caused by the high interparticle stiffness that hinders compression. The reverse observation holds for the tangential component of the relative displacement vector. The contribution of the deformation of the empty space between physical contacts to the continuum, macro-scale strain tensor is therefore very important for the understanding and the prediction of the macro-scale deformation of granular materials.

Analysis of three-dimensional micro-mechanical strain formulations for granular materials: Evaluation of accuracy

An important objective of recent research on micro-mechanics of granular materials is to develop macroscopic constitutive relations in terms of micro-mechanical quantities at inter-particle contacts. Although the micro-mechanical formulation of the stress tensor is well established, the corresponding formulation for the strain tensor has proven to be much more evasive, still being the subject of much discussion. In this paper, we study various micro-mechanical strain formulations for three-dimensional granular assemblies, following the work of Bagi in two dimensions ( Bagi, 2006). All of these formulations are either based on an equivalent continuum approach, or follow the best-fit approach. Their accuracy is evaluated by comparing their results, using data from Discrete Element Method simulations on periodic assemblies, to the macroscopic deformation. It is found that Bagi’s formulation ( Bagi, 1996), which is based on the Delaunay tessellation of space, is the most accurate. Furthermore, the best-fit formulation based on particle displacements only did unexpectedly well, in contrast to previously reported results for two-dimensional assemblies.

New Perspectives in Stochastic Geometry

The principal aim of stochastic geometry is the mathematical analysis of random geometric structures. Fundamental examples of such structures are point processes of geometric objects, random tessellations of space into convex or non-convex regions, random systems of non-overlapping balls (or more general convex bodies), or excursion and level sets of Gaussian random fields. The workshop was devoted to the discussion and exploration of recent advances in stochastic geometry and its related areas. Among the 23 participants were many leading figures in the field as well as some very promising young scientists. One main theme of the workshop concerned new results and methodology in classical stochastic geometry. In recent years it has become possible not only to make conclusions about mean values of geometric quantities, but also to derive distributional properties, to prove limit theorems and large deviation results, and to explore higher-order moment properties. Examples include: Hug, Reitzner (approximation of polytopes), Calka (visibility properties of the Boolean model), Schreiber (polygonal Markov fields), Penrose (normal approximation in random geometric graphs), Yukich (limit theorems), Reitzner (random tessellations), Baccelli (point processes and information theory). A second main theme of the workshop concerned allocation and matching procedures for point processes and random measures.The study of such allocations has resulted in some remarkable progress in the understanding of invariance properties of Palm probability measures and associated transport and coupling questions.There are deep and striking relationships with random tessellations, classical optimal transport theory, potential theory, and complex analysis. Examples include: Peres (fair allocations, optimal matchings, gravitational allocation, ...), Sturm (optimal transportation of measures on metric spaces), Thorisson (transport kernels and mass-stationarity), Last (Cox and Bernoulli transports). Three series of lectures formed the organizational backbone of the workshop. They were delivered by Matthias Reitzner on "Recent results in stochastic geometry", by Yuval Peres on "Fair allocations" and by Theo Sturm on "New trends in optimal transportation on Riemannian and singular spaces". Many of the contributed talks were related to one or more issues covered by these lectures. Another integral part of the programme was provided by three discussion sessions "Probability models with geometric flavour" organized by Jesper Møller and Sergei Zouev, "Asymptotics in stochastic geometry" organized by Peter Mörters and Mathew Penrose, and "From convex to metric and fractal geometry" organized by Martina Zähle and Wolfgang Weil. These sessions provided space for participants to bring up new ideas and discuss open problems in an informal manner. Many participants of the workshop co-authored a collection of papers "New Perspectives in Stochastic Geometry" edited by W.S. Kendall and I. Molchanov that will be published by the Oxford University Press in 2009.

A solution to the Path Planning problem using angle preprocessing

The Path Planning problem is a common topic for Robotics and Computational Geometry. Many important results have been found to this classic problem, some of them based on plane or space tessellation. The new approach we propose in this paper computes a partition of the plane called the Polar Diagram, using angle properties as criterion of construction. Compared to some other plane partitions as Voronoi Diagrams, this tessellation can be computed much more efficiently for different geometric objects. The polar diagram used as preprocessing can be applied to many geometric problems where the solution can be given by angle processing, such as Visibility or Path Planning problems.

The Representation and Matching of Images Using Top Points

In previous work, singular points (or top points) in the scale space representation of generic images have proven valuable for image matching. In this paper, we propose a novel construction that encodes the scale space description of top points in the form of a directed acyclic graph. This representation allows us to utilize coarse-to-fine graph matching algorithms for comparing images represented in terms of top point configurations instead of using solely the top points and their features in a point matching algorithm, as was done previously. The nodes of the graph represent the critical paths together with their top points. The edge set captures the neighborhood distribution of vertices in scale space, and is constructed through a hierarchical tessellation of scale space using a Delaunay triangulation of the top points. We present a coarse-to-fine many-to-many matching algorithm for comparing such graph-based representations. The algorithm is based on a metric-tree representation of labeled graphs and their low-distortion embeddings into normed vector spaces via spherical encoding. This is a two-step transformation that reduces the matching problem to that of computing a distribution-based distance measure between two such embeddings. To evaluate the quality of our representation, four sets of experiments are performed. First, the stability of this representation under Gaussian noise of increasing magnitude is examined. Second, a series of recognition experiments is run on a face database. Third, a set of clutter and occlusion experiments is performed to measure the robustness of the algorithm. Fourth, the algorithm is compared to a leading interest point-based framework in an object recognition experiment.

Structure and Stability of Hydrogen Clathrates of Ammonia Borane

AbstractStructural rules have been formulated for the molecular crystal of ammonia borane (AB, NH3BH3) and clathrates thereof. These rules are similar to the “ice rules” in water. Periodic structures of possible clathrates of AB have been predicted. A rigorous analysis of the uniform space filling tessellations of 3D space has been performed resulting in the identification of two promising structures for AB clathrates. Additional five structures have been identified exploiting features and properties of AB molecules. The screening, or evaluation, of proposed periodic structures has been performed on the basis of their stability determined at the density functional level of theory. Hydrogen capacity of the most stable periodic structure (cantitruncated cubic honeycomb) is estimated to be 21 wt%, 19% chemically bound in AB and 2 wt% of H2 physisorbed in the cages of AB.

Cell size distribution in a random tessellation of space governed by the Kolmogorov-Johnson-Mehl-Avrami model: Grain size distribution in crystallization

The space subdivision in cells resulting from a process of random nucleation and growth is a subject of interest in many scientific fields. In this paper, we deduce the expected value and variance of these distributions while assuming that the space subdivision process is in accordance with the premises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions on the time dependency of nucleation and growth rates. We have also developed an approximate analytical cell size probability density function. Finally, we have applied our approach to the distributions resulting from solid phase crystallization under isochronal heating conditions.

Partitioning 3-homogeneous latin bitrades

A latin bitrade \({(T^{\diamond},\, T^{\otimes})}\) is a pair of partial latin squares that define the difference between two arbitrary latin squares \({L^{\diamond} \supseteq T^{\diamond}}\) and \({L^{\otimes} \supseteq T^{\otimes}}\) of the same order. A 3-homogeneous bitrade \({(T^{\diamond},\, T^{\otimes})}\) has three entries in each row, three entries in each column, and each symbol appears three times in \({T^{\diamond}}\). Cavenagh [2] showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.

Erratum: Cell size distribution in random tessellations of space [Phys. Rev. E70, 066119 (2004)

Erratum: Cell size distribution in random tessellations of space [Phys. Rev. E70, 066119 (2004)

PREDICTION OF THE POSITION OF AN ANIMAL BASED ON POPULATIONS OF GRID AND PLACE CELLS: A COMPARATIVE SIMULATION STUDY

The grid cells of the rodent medial entorhinal cortex (MEC) show activity patterns correlated with the animal's position. Unlike hippocampal place cells that are activated at only one specific location in the environment, MEC grid cells increase firing frequency at multiple regions in space, or subfields, that are arranged in regular triangular grids. It has been recently shown that a conjunction of MEC grid cells can lead to unique spatial representations. However, it remains unclear what the key properties of the grids are that allow for an accurate reconstruction of the position of the animal and what the comparison with hippocampal place cells is. Here we use a theoretical approach based on data from electrophysiological recordings of the MEC to simulate the neural activity of grid cells. Our simulations account for the accurate reproduction of grid cell mean firing rates, based on only three grid parameters, that is grid phase, spacing and orientation. The analysis of the key properties of the grids first reveals that for an accurate position reconstruction, it is necessary to combine cells with different grid spacings (which are found at different dorsoventral locations of the MEC) or orientations. Second, the relationship between grid spacing and subfield size observed in physiological data is optimal to predict the animal's position. Third, the regular triangular tessellating patterns of grid cells lead to the best position reconstruction results when compared with all other regular tessellations of two-dimensional space. Finally, the comparison of grid cells with place cells shows that populations of MEC grid cells can better predict the animal's position than equally-sized populations of hippocampal place cells with similar but unique spatial fields. Taken together, our results suggest that the MEC provides highly compact representations of the animal's position, which may be subsequently integrated by the place cells of the hippocampus.

On the size distribution of Poisson Voronoi cells

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two- and three-dimensional (3D) spaces there is no exact result known for the size distribution of Voronoi cells. Motivated by the simple form of the distribution function in the 1D case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size-distribution function in the practically important 2D and 3D cases as well. Denoting the dimensionality of the space by d ( d = 1 , 2 , 3 ) the f ( y ) = Const * y ( 3 d - 1 ) / 2 exp ( - ( 3 d + 1 ) y / 2 ) compact form is suggested for the normalized cell-size distribution function. By using large-scale computer simulations the viability of the proposed distribution function is studied and critically discussed.

Atomic-scale simulation of martensitic phase transformations in NiAl

Thermodynamic and kinetic properties of the martensitic phase transformations (MPT) in a model NiAl alloy are studied using the molecular dynamics method. The interatomic potential of Farkas et al. is used. It allows simulating a quite realistic behavior of the MPT for this alloy. We observe a typical hysteretic behavior during the reversible MPT. The temperature and composition regimes appear to be in good agreement with the ones experimentally observed for the Ni–Al alloy system. A new way of characterizing the local structure is proposed and applied for the analysis of the phase transformation. The definition of the local order parameter is based on the technique of the common neighbor analysis and combined with the method of space tessellation in Voronoy polyhedra. With this parameter the phase transformation can be followed in time on an atomistic scale showing how the nucleation process and the volume fraction of the new phase evolve with time. In order to find the location of the MPT in the temperature–composition–stress diagram, the influence of shear stresses is investigated. The effects of grain boundaries are simulated.