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Tilings
Terms modified by Tilings Selected AbstractsTiling among stereotyped dendritic branches in an identified Drosophila motoneuronTHE JOURNAL OF COMPARATIVE NEUROLOGY, Issue 12 2010F. Vonhoff Shape similarities of an identified Drosophila motoneuron across animals. Overlay of three geometric dendrite reconstructions of a Drosophila flight motoneuron, based on intracellular fills of this motoneuron in different animals with identical genotypes. Although similarities exist in the overall branching structure, and the dendritic territories are nearly identical in all three cells, marked differences exist in the fine branching structure. The Journal of Comparative Neurology, Volume 518, Number 12, pages 2169,2185. [source] Tiling among stereotyped dendritic branches in an identified Drosophila motoneuron,,THE JOURNAL OF COMPARATIVE NEUROLOGY, Issue 12 2010F. Vonhoff Abstract Different types of neurons can be distinguished by the specific targeting locations and branching patterns of their dendrites, which form the blueprint for wiring the brain. Unraveling which specific signals control different aspects of dendritic architecture, such as branching and elongation, pruning and cessation of growth, territory formation, tiling, and self-avoidance requires a quantitative comparison in control and genetically manipulated neurons. The highly conserved shapes of individually identified Drosophila neurons make them well suited for the analysis of dendritic architecture principles. However, to date it remains unclear how tightly dendritic architecture principles of identified central neurons are regulated. This study uses quantitative reconstructions of dendritic architecture of an identified Drosophila flight motoneuron (MN5) with a complex dendritic tree, comprising more than 4,000 dendritic branches and 6 mm total length. MN5 contains a fixed number of 23 dendritic subtrees, which tile into distinct, nonoverlapping volumes of the diffuse motor neuropil. Across-animal comparison and quantitative analysis suggest that tiling of the different dendritic subtrees of the same neuron is caused by competitive and repulsive interactions among subtrees, perhaps allowing different dendritic compartments to be connected to different circuit elements. We also show that dendritic architecture is similar among different wildtype and GAL4 driver fly lines. Metric and topological dendritic architecture features are sufficiently constant to allow for studies of the underlying control mechanisms by genetic manipulations. Dendritic territory and certain topological measures, such as tree compactness, are most constant, suggesting that these reflect the intrinsic molecular identity of the neuron. J. Comp. Neurol. 518:2169,2185, 2010. © 2010 Wiley-Liss, Inc. [source] Designing materials with prescribed elastic properties using polygonal cellsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2003Alejandro R. Diaz Abstract An extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional effort. In particular, the periodic homogenization problem defined on polygonal base cells that tile the plane can be replaced and analysed more efficiently by an equivalent problem that uses simple parallelograms as base cells. Different material layouts can be obtained by varying just two parameters that affect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an efficient way to organize the search of the design space for all possible single-scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results. Copyright © 2003 John Wiley & Sons, Ltd. [source] CHARACTERIZATION OF FOOD SURFACES USING SCALE-SENSITIVE FRACTAL ANALYSISJOURNAL OF FOOD PROCESS ENGINEERING, Issue 2 2000FRANCO PEDRESCHI ABSTRACT Length-scale and area-scale analyses, two of the scale-sensitive fractal analyses performed by the software Surfraxhttp://www.surfract.com, were used to study food surfaces measured with a scanning laser microscope (SLM). The SLM measures surfaces, or textures (i.e., acquires topographical data as a collection of heights as a function of position), at a spatial and vertical resolution of 25 ,m. The measured textures are analyzed by using linear and areal tiling (length-scale and area-scale analysis) and by conventional statistical analyses. Area-scale and length-scale fractal complexities (Lsfc and Asfc) and the smooth-rough crossover (SRC) are derived from the scale-sensitive fractal analyses. Both measures proved adequate to quantify and differentiate surfaces of foods (e.g., chocolate and a slice of bread), which were smooth or porous to the naked eye. Surfaces generated after frying of potato products (e.g., potato chips and French fries) had similar values of Asfc and SRC, and larger (implying more complex and rougher surfaces) than those of the raw potato. Variability of surface texture characterization parameters as a function of the size of the measured region was used in selecting the size of the measured regions for further analysis. The length-scale method of profile analysis (also called the Richardson or compass method) was useful in determining the directionality or lay of the anisotropic texture on food surfaces. [source] Tilings, packings, coverings, and the approximation of functionsMATHEMATISCHE NACHRICHTEN, Issue 1 2004Aicke Hinrichs Abstract A packing (resp. covering) , of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ,. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real-valued functions that rests on so-called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non-compact spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Bandgap properties of low-index contrast aperiodically ordered photonic quasicrystalsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 11 2009Gianluigi Zito Abstract We numerically analyze, using Finite Difference Time Domain simulations, the bandgap properties of photonic quasicrystals with a low-index contrast. We compared 8-, 10-, and 12-fold symmetry aperiodically ordered lattices with different spatial tiling. Our results show that tiling design, more than symmetry, determines the transmission properties of these structures. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2732,2737, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24724 [source] Minimum boundary touching tilings of polyominoesMLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2006Andreas Spillner Abstract We study the problem of tiling a polyomino P with as few squares as possible such that every square in the tiling has a non-empty intersection with the boundary of P . Our main result is an algorithm which given a simply connected polyomino P computes such a tiling of P . We indicate how one can improve the running time of this algorithm for the more restricted row-column-convex polyominoes. Finally we show that a related decision problem is in NP for rectangular polyominoes. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Space fullerenes: a computer search for new Frank,Kasper structuresACTA CRYSTALLOGRAPHICA SECTION A, Issue 5 2010Mathieu Dutour Sikiri A Frank,Kasper structure is a 3-periodic tiling of the Euclidean space by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank,Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank,Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank,Kasper,Sullivan, Shoemaker,Shoemaker, Sadoc,Mosseri and Deza,Shtogrin. [source] Structure factor for decorated Penrose tiling in physical spaceACTA CRYSTALLOGRAPHICA SECTION A, Issue 4 2010omiej Kozakowski The structure factor for an arbitrarily decorated Penrose tiling has been calculated in the average unit cell description. The obtained formula uses only the physical coordinates of the atoms decorating a structure. The final equation can be easily extended so that it can describe the other physical properties of a structure. Its usefulness is demonstrated by its use in the Al,Ni,Co alloy structure-refinement process. [source] Dense quasicrystalline tilings by squares and equilateral trianglesACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2010Michael O'Keeffe Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses. [source] Point substitution processes for decagonal quasiperiodic tilingsACTA CRYSTALLOGRAPHICA SECTION A, Issue 5 2009Nobuhisa Fujita A general construction principle for the inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of expansion and division of the tiles, where the expanded tiles can be divided arbitrarily as long as the set of prototiles is maintained. A certain kind of point decoration process turns out to be useful for the identification of possible division rules. The method is capable of generating a broad range of decagonal tilings, many of which are chiral and have atomic surfaces with fractal boundaries. Two new families of decagonal tilings are presented; one is quaternary and the other ternary. The properties of the ternary tilings with rhombic, pentagonal and hexagonal prototiles are investigated in detail. [source] Quasiperiodic plane tilings based on stepped surfacesACTA CRYSTALLOGRAPHICA SECTION A, Issue 3 2008A. V. Shutov Static and dynamic characteristics of layerwise growth in two-dimensional quasiperiodic Ito,Ohtsuki tilings are studied. These tilings are the projections of three-dimensional stepped surfaces. It is proved that these tilings have hexagonal self-similar growth with bounded radius of neighborhood. A formula is given for the averaged coordination number. Deviations of coordination numbers from its average are quasiperiodic. Ito,Ohtsuki tiling can be decomposed into one-dimensional sector layers. These sector layers are one-dimensional quasiperiodic tilings with properties like Ito,Ohtsuki tilings. [source] Random dyadic tilings of the unit squareRANDOM STRUCTURES AND ALGORITHMS, Issue 3-4 2002Svante Janson A "dyadic rectangle" is a set of the form R = [a2,s, (a + 1)2,s] × [b2,t, (b + 1)2,t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n -tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2,n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n -tilings, and study some limiting properties of random tilings. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 225,251, 2002 [source] Combinatorial construction of tilings by barycentric simplex orbits (D symbols) and their realizations in Euclidean and other homogeneous spacesACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2005Emil Molnár A new method, developed in previous works by the author (partly with co-authors), is presented which decides algorithmically, in principle by computer, whether a combinatorial space tiling (,,) is realizable in the d -dimensional Euclidean space Ed (think of d = 2, 3, 4) or in other homogeneous spaces, e.g. in Thurston's 3-geometries: Then our group , will be an isometry group of a projective metric 3-sphere , acting discontinuously on its above tiling . The method is illustrated by a plane example and by the well known rhombohedron tiling , where , = Rm is the Euclidean space group No. 166 in International Tables for Crystallography. [source] Morphology and mosaics of melanopsin-expressing retinal ganglion cell types in miceTHE JOURNAL OF COMPARATIVE NEUROLOGY, Issue 13 2010David M. Berson Abstract Melanopsin is the photopigment of intrinsically photosensitive retinal ganglion cells (ipRGCs). Melanopsin immunoreactivity reveals two dendritic plexuses within the inner plexiform layer (IPL) and morphologically heterogeneous retinal ganglion cells. Using enhanced immunohistochemistry, we provide a fuller description of murine cell types expressing melanopsin, their contribution to the plexuses of melanopsin dendrites, and mosaics formed by each type. M1 cells, corresponding to the originally described ganglion-cell photoreceptors, occupy the ganglion cell or inner nuclear layers. Their large, sparsely branched arbors (mean diameter 275 ,m) monostratify at the outer limit of the OFF sublayer. M2 cells also have large, monostratified dendritic arbors (mean diameter 310 ,m), but ramify in the inner third of the IPL, within the ON sublayer. There are ,900 M1 cells and 800 M2 cells per retina; each type comprises roughly 1,2% of all ganglion cells. The cell bodies of M1 cells are slightly smaller than those of M2 cells (mean diameters: 13 ,m for M1, 15 ,m for M2). Dendritic field overlap is extensive within each type (coverage factors ,3.8 for M1 and 2.5 for M2 cells). Rare bistratified cells deploy terminal dendrites within both melanopsin-immunoreactive plexuses. Because these are too sparsely distributed to permit complete retinal tiling, they lack a key feature of true ganglion cell types and may be anomalous hybrids of the M1 and M2 types. Finally, we observed weak melanopsin immunoreactivity in other ganglion cells, mostly with large somata, that may constitute one or more additional types of melanopsin-expressing cells. J. Comp. Neurol. 518:2405,2422, 2010. © 2010 Wiley-Liss, Inc. [source] Morphology and mosaics of melanopsin-expressing retinal ganglion cell types in mice,THE JOURNAL OF COMPARATIVE NEUROLOGY, Issue 13 2010David M. Berson Abstract Melanopsin is the photopigment of intrinsically photosensitive retinal ganglion cells (ipRGCs). Melanopsin immunoreactivity reveals two dendritic plexuses within the inner plexiform layer (IPL) and morphologically heterogeneous retinal ganglion cells. Using enhanced immunohistochemistry, we provide a fuller description of murine cell types expressing melanopsin, their contribution to the plexuses of melanopsin dendrites, and mosaics formed by each type. M1 cells, corresponding to the originally described ganglion-cell photoreceptors, occupy the ganglion cell or inner nuclear layers. Their large, sparsely branched arbors (mean diameter 275 ,m) monostratify at the outer limit of the OFF sublayer. M2 cells also have large, monostratified dendritic arbors (mean diameter 310 ,m), but ramify in the inner third of the IPL, within the ON sublayer. There are ,900 M1 cells and 800 M2 cells per retina; each type comprises roughly 1,2% of all ganglion cells. The cell bodies of M1 cells are slightly smaller than those of M2 cells (mean diameters: 13 ,m for M1, 15 ,m for M2). Dendritic field overlap is extensive within each type (coverage factors ,3.8 for M1 and 4.6 for M2 cells). Rare bistratified cells deploy terminal dendrites within both melanopsin-immunoreactive plexuses. Because these are too sparsely distributed to permit complete retinal tiling, they lack a key feature of true ganglion cell types and may be anomalous hybrids of the M1 and M2 types. Finally, we observed weak melanopsin immunoreactivity in other ganglion cells, mostly with large somata, that may constitute one or more additional types of melanopsin-expressing cells. J. Comp. Neurol. 518:2405,2422, 2010. © 2010 Wiley-Liss, Inc. [source] Tiling among stereotyped dendritic branches in an identified Drosophila motoneuron,,THE JOURNAL OF COMPARATIVE NEUROLOGY, Issue 12 2010F. Vonhoff Abstract Different types of neurons can be distinguished by the specific targeting locations and branching patterns of their dendrites, which form the blueprint for wiring the brain. Unraveling which specific signals control different aspects of dendritic architecture, such as branching and elongation, pruning and cessation of growth, territory formation, tiling, and self-avoidance requires a quantitative comparison in control and genetically manipulated neurons. The highly conserved shapes of individually identified Drosophila neurons make them well suited for the analysis of dendritic architecture principles. However, to date it remains unclear how tightly dendritic architecture principles of identified central neurons are regulated. This study uses quantitative reconstructions of dendritic architecture of an identified Drosophila flight motoneuron (MN5) with a complex dendritic tree, comprising more than 4,000 dendritic branches and 6 mm total length. MN5 contains a fixed number of 23 dendritic subtrees, which tile into distinct, nonoverlapping volumes of the diffuse motor neuropil. Across-animal comparison and quantitative analysis suggest that tiling of the different dendritic subtrees of the same neuron is caused by competitive and repulsive interactions among subtrees, perhaps allowing different dendritic compartments to be connected to different circuit elements. We also show that dendritic architecture is similar among different wildtype and GAL4 driver fly lines. Metric and topological dendritic architecture features are sufficiently constant to allow for studies of the underlying control mechanisms by genetic manipulations. Dendritic territory and certain topological measures, such as tree compactness, are most constant, suggesting that these reflect the intrinsic molecular identity of the neuron. J. Comp. Neurol. 518:2169,2185, 2010. © 2010 Wiley-Liss, Inc. [source] A simple isohedral tiling of three-dimensional space by infinite tiles and with symmetry IadACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2002O. Delgado Friedrichs A tiling of space by tiles that have all hexagonal faces and are infinite in one direction is described. The tiling is simple (four tiles meet at each vertex, three at each edge and two at each face) and carries a 4-connected net whose vertices are the lattice complex S* with symmetry Iad. The tiling is closely related to the densest cubic cylinder packing, ,. It is shown that the other invariant cubic lattice complexes unique to Iad (Y** and V*) are also related to the same cylinder packing. [source] Analysis of ion-migration paths in inorganic frameworks by means of tilings and Voronoi,Dirichlet partition: a comparisonACTA CRYSTALLOGRAPHICA SECTION B, Issue 4 2009Nataly A. Anurova Two methods using Voronoi,Dirichlet polyhedra (Voronoi,Dirichlet partition) or tiles (tiling) based on partitioning space are compared to investigate cavities and channels in crystal structures. The tiling method was applied for the first time to study ion conductivity in 105 ternary, lithium,oxygen-containing compounds, LiaXbOz, that were recently recognized as fast-ion conductors with the Voronoi,Dirichlet partition method. The two methods were found to be similar in predicting the occurrence of ionic conductivity, however, their conclusions on the dimensionality of conductivity were different in two cases. It is shown that such a contradiction can indicate a high anisotropy of conductivity. Both advantages and restrictions of the methods are discussed with respect to fast-ion conductors and zeolites. [source] Design of a Molecular QuasicrystalCHEMPHYSCHEM, Issue 8 2006Zhongfu Zhou Dr. Designer materials: The authors propose a design strategy for a quasicrystalline material composed of discrete molecular entities. The molecular quasicrystal (see figure), which is based on a standard Penrose tiling, is energetically stable and gives rise to a 10-fold symmetric diffraction pattern. The strategy may be further exploited to design molecular quasicrystals based on a range of different types of quasiperiodic arrays. [source] Brane tilings and their applicationsFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 6 2008M. Yamazaki Abstract We review recent developments in the theory of brane tilings and four-dimensional ,, = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers and some works in progress [1,7]. [source] On the discretization of problems involving periodic planar tilingsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2001André Bénard Abstract Features related to the discretization of problems characterized by simple periodic tilings using cells of various shapes are discussed. Various cell geometries that tile the plane periodically are considered. Equivalent problems are identified, where the discretization can take place on a parallelogram, regardless of the shape of the original cell. These equivalent problems also suggest a numbering of the equations that results in matrices with interesting and useful properties. Copyright © 2001 John Wiley & Sons, Ltd. [source] Minimum boundary touching tilings of polyominoesMLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2006Andreas Spillner Abstract We study the problem of tiling a polyomino P with as few squares as possible such that every square in the tiling has a non-empty intersection with the boundary of P . Our main result is an algorithm which given a simply connected polyomino P computes such a tiling of P . We indicate how one can improve the running time of this algorithm for the more restricted row-column-convex polyominoes. Finally we show that a related decision problem is in NP for rectangular polyominoes. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetryACTA CRYSTALLOGRAPHICA SECTION A, Issue 3 2010A. V. Shutov A class of quasiperiodic tilings of the complex plane is discussed. These tilings are based on ,-expansions corresponding to cubic irrationalities. There are three classes of tilings: Q3, Q4 and Q5. These classes consist of three, four and five pairwise similar prototiles, respectively. A simple algorithm for construction of these tilings is considered. This algorithm uses greedy expansions of natural numbers on some sequence. Weak and strong parameterizations for tilings are obtained. Layerwise growth, the complexity function and the structure of fractal boundaries of tilings are studied. The parameterization of vertices and boundaries of tilings, and also similarity transformations of tilings, are considered. [source] Dense quasicrystalline tilings by squares and equilateral trianglesACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2010Michael O'Keeffe Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses. [source] Point substitution processes for decagonal quasiperiodic tilingsACTA CRYSTALLOGRAPHICA SECTION A, Issue 5 2009Nobuhisa Fujita A general construction principle for the inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of expansion and division of the tiles, where the expanded tiles can be divided arbitrarily as long as the set of prototiles is maintained. A certain kind of point decoration process turns out to be useful for the identification of possible division rules. The method is capable of generating a broad range of decagonal tilings, many of which are chiral and have atomic surfaces with fractal boundaries. Two new families of decagonal tilings are presented; one is quaternary and the other ternary. The properties of the ternary tilings with rhombic, pentagonal and hexagonal prototiles are investigated in detail. [source] Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examplesACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2009S. J. Ramsden We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G). [source] Quasiperiodic plane tilings based on stepped surfacesACTA CRYSTALLOGRAPHICA SECTION A, Issue 3 2008A. V. Shutov Static and dynamic characteristics of layerwise growth in two-dimensional quasiperiodic Ito,Ohtsuki tilings are studied. These tilings are the projections of three-dimensional stepped surfaces. It is proved that these tilings have hexagonal self-similar growth with bounded radius of neighborhood. A formula is given for the averaged coordination number. Deviations of coordination numbers from its average are quasiperiodic. Ito,Ohtsuki tiling can be decomposed into one-dimensional sector layers. These sector layers are one-dimensional quasiperiodic tilings with properties like Ito,Ohtsuki tilings. [source] Random dyadic tilings of the unit squareRANDOM STRUCTURES AND ALGORITHMS, Issue 3-4 2002Svante Janson A "dyadic rectangle" is a set of the form R = [a2,s, (a + 1)2,s] × [b2,t, (b + 1)2,t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n -tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2,n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n -tilings, and study some limiting properties of random tilings. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 225,251, 2002 [source] Combinatorial construction of tilings by barycentric simplex orbits (D symbols) and their realizations in Euclidean and other homogeneous spacesACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2005Emil Molnár A new method, developed in previous works by the author (partly with co-authors), is presented which decides algorithmically, in principle by computer, whether a combinatorial space tiling (,,) is realizable in the d -dimensional Euclidean space Ed (think of d = 2, 3, 4) or in other homogeneous spaces, e.g. in Thurston's 3-geometries: Then our group , will be an isometry group of a projective metric 3-sphere , acting discontinuously on its above tiling . The method is illustrated by a plane example and by the well known rhombohedron tiling , where , = Rm is the Euclidean space group No. 166 in International Tables for Crystallography. [source] | ||||||||||||||||