To see the other types of publications on this topic, follow the link: Tessellation.
Journal articles on the topic 'Tessellation'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 journal articles for your research on the topic 'Tessellation.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.
1
Chang, Wei. "Application of Tessellation in Architectural Geometry Design." E3S Web of Conferences 38 (2018): 03015. http://dx.doi.org/10.1051/e3sconf/20183803015.
Full textAbstract:
Tessellation plays a significant role in architectural geometry design, which is widely used both through history of architecture and in modern architectural design with the help of computer technology. Tessellation has been found since the birth of civilization. In terms of dimensions, there are two- dimensional tessellations and three-dimensional tessellations; in terms of symmetry, there are periodic tessellations and aperiodic tessellations. Besides, some special types of tessellations such as Voronoi Tessellation and Delaunay Triangles are also included. Both Geometry and Crystallography, the latter of which is the basic theory of three-dimensional tessellations, need to be studied. In history, tessellation was applied into skins or decorations in architecture. The development of Computer technology enables tessellation to be more powerful, as seen in surface control, surface display and structure design, etc. Therefore, research on the application of tessellation in architectural geometry design is of great necessity in architecture studies.
2
Nguyen, Linh Ngoc, Viola Weiss, and Richard Cowan. "COLUMN TESSELLATIONS." Image Analysis & Stereology 34, no. 2 (June 28, 2015): 87. http://dx.doi.org/10.5566/ias.1285.
Full textAbstract:
A new class of non facet-to-facet random tessellations in three-dimensional space is introduced -- the so-called column tessellations. The spatial construction is based on a stationary planar tessellation; each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities, topological and metric mean values of the spatial tessellation can be calculated from suitably chosen parameters of the planar tessellation.
3
Nagel, Werner, and Viola Weiss. "Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration." Advances in Applied Probability 37, no. 4 (December 2005): 859–83. http://dx.doi.org/10.1239/aap/1134587744.
Full textAbstract:
Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.
4
Nagel, Werner, and Viola Weiss. "Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration." Advances in Applied Probability 37, no. 04 (December 2005): 859–83. http://dx.doi.org/10.1017/s0001867800000574.
Full textAbstract:
Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.
5
Blackwell, Paul G., and Jesper Møller. "Bayesian analysis of deformed tessellation models." Advances in Applied Probability 35, no. 1 (March 2003): 4–26. http://dx.doi.org/10.1239/aap/1046366096.
Full textAbstract:
We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.
6
Blackwell, Paul G., and Jesper Møller. "Bayesian analysis of deformed tessellation models." Advances in Applied Probability 35, no. 01 (March 2003): 4–26. http://dx.doi.org/10.1017/s0001867800012052.
Full textAbstract:
We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.
7
Mecke, Joseph, Werner Nagel, and Viola Weiss. "The iteration of random tessellations and a construction of a homogeneous process of cell divisions." Advances in Applied Probability 40, no. 1 (March 2008): 49–59. http://dx.doi.org/10.1239/aap/1208358886.
Full textAbstract:
A random tessellation of ℝd is said to be homogeneous if its distribution is invariant under all shifts of ℝd. The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.
8
Mecke, Joseph, Werner Nagel, and Viola Weiss. "The iteration of random tessellations and a construction of a homogeneous process of cell divisions." Advances in Applied Probability 40, no. 01 (March 2008): 49–59. http://dx.doi.org/10.1017/s0001867800002378.
Full textAbstract:
A random tessellation of ℝ d is said to be homogeneous if its distribution is invariant under all shifts of ℝ d . The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.
9
Kizilörenli, Ecenur, and Feray Maden. "Tessellation in Architecture from Past to Present." IOP Conference Series: Materials Science and Engineering 1203, no. 3 (November 1, 2021): 032062. http://dx.doi.org/10.1088/1757-899x/1203/3/032062.
Full textAbstract:
Abstract Tessellation, which has examples of use in art and architecture, is the covering of a surface using one or more geometric shapes without overlapping or gaps. Based on Roman mosaics, the tessellation has an important place in architecture since the ancient times. Through the history, different patterns have been used by many cultures for various applications ranging from decorative covering elements to multi-functional latticework screens. The tessellation has still been used in contemporary architecture since it not only allows creating the geometrical surface in an order but also provides multi-functionality to the surface when applied as shading elements. The tessellation can be reviewed under three categories such as regular, semi-regular and demi-regular tessellations. Two- and three-dimensional examples of the tessellations can be seen in contemporary architecture either as façade elements or patterns used for structural elements. Because the tessellation plays a significant role in architecture in terms of geometrical or structural design, the interest on this topic has been increased in recent years. Due to their great potentials, more studies should be conducted on the tessellations. For this reason, within the scope of this paper, the applied examples of the tessellations in buildings from past to present are examined which include both static and kinetic ones. In this paper, the geometric design principles, combination methods and iteration processes of the examples are also presented. As well as providing a deeper understanding of such tessellation methods, this study will serve as a basis of reference for future studies in this field.
10
Ponížil, Petr, Ivan Saxl, and Jaroslav Procházka. "Classification and Computer Simulation of 2D Tessellations." Materials Science Forum 567-568 (December 2007): 281–84. http://dx.doi.org/10.4028/www.scientific.net/msf.567-568.281.
Full textAbstract:
In an analogy to the 3D tool of tessellation classification – w−s diagram, a similar graphical device is proposed for 2D tessellations. Any tessellation is represented by a point in the Cartesian coordinate system with the axes Ep (the mean cell perimeter) and CV a (the coefficient of cell area variation). Images of tessellations and p−CV a diagrams for selected tessellations with low and high values of CV a are shown as examples.
11
Sadahiro, Yukio. "An Exploratory Method for Analyzing a Spatial Tessellation in Relation to a Set of other Spatial Tessellations." Environment and Planning A: Economy and Space 34, no. 6 (June 2002): 1037–58. http://dx.doi.org/10.1068/a34230.
Full textAbstract:
Spatial tessellation is one of the most important spatial structures in geography. There are various types of spatial tessellations such as administrative units, school districts, and census tracts. Spatial tessellations are often closely related to each other; school districts are determined by, say, administrative units and land uses; electoral districts are related to administrative units, local communities, census tracts, and so forth. Such relationships among spatial tessellations have drawn the attention of geographers; to what extent is the development of a spatial tessellation affected by a set of other tessellations? To give a clue to the answer to this question, in this paper I propose three methods for analyzing a spatial tessellation in relation to a set of other tessellations: the region-based method, the boundary-based method, and the hybrid method. They are all designed for exploratory spatial analysis rather than confirmatory analysis. The methods are evaluated through an empirical study, analysis of the administrative system in Ponneri, India, in the late 18th century.
12
Lautensack, Claudia, and Sergei Zuyev. "Random Laguerre tessellations." Advances in Applied Probability 40, no. 3 (September 2008): 630–50. http://dx.doi.org/10.1239/aap/1222868179.
Full textAbstract:
A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.
13
Lautensack, Claudia, and Sergei Zuyev. "Random Laguerre tessellations." Advances in Applied Probability 40, no. 03 (September 2008): 630–50. http://dx.doi.org/10.1017/s000186780000272x.
Full textAbstract:
A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.
14
Chun, Junho, and Zai Shi. "Algorithmic analysis and application of structural tessellation in design and optimization." MATEC Web of Conferences 396 (2024): 05008. http://dx.doi.org/10.1051/matecconf/202439605008.
Full textAbstract:
This paper explores the application of structural tessellation in architectural and structural design, where surfaces or spaces are divided into smaller repeating shapes or units to create aesthetic and functional structures. Structural tessellation offers various advantages, including improved stability, increased load-bearing capacity, and enhanced aesthetic appeal. The growing use of digital tools and advanced numerical algorithms has facilitated the creation of complex and intricate tessellations that can be tailored to suit specific project requirements. This research focuses on algorithmic methods for generating tessellations and their utilization in structural engineering and design optimization. Diverse patterns and configurations of tessellation are investigated through mesh generation algorithms, parametric approaches, and pattern gradation and repetitions, with data visualization accomplished using computing scripts and Grasshopper. The tessellation elements are employed to discretize the design domain in structural optimization and create initial patterns. The paper demonstrates the feasibility of proposed frameworks for structural design and application through the examination of various numerical examples. In conclusion, the strategic use of structural tessellation proves to be effective in producing unique and functional structures that seamlessly combine visual appeal with material and resource efficiency.
15
Cowan, Richard, and Christoph Thäle. "THE CHARACTER OF PLANAR TESSELLATIONS WHICH ARE NOT SIDE-TO-SIDE." Image Analysis & Stereology 33, no. 1 (March 5, 2014): 39. http://dx.doi.org/10.5566/ias.v33.p39-54.
Full textAbstract:
This paper studies stationary tessellations and tilings of the plane in which all cells are convex polygons. The focus is on the class of tessellations which are not side-to-side. The character of these tessellations is explored, with special attention paid to the relationship between edges of the tessellation and sides of the polygonal cells and to the combinatorial topology between the ‘adjacent’ geometric elements of the tessellation. Three new parameters, e0,e1 and e2 summing to unity, are introduced. These capture the essence of non side-to-side tessellations and play a role in understanding the adjacency of sides and cells. Examples illustrate the theory.
16
Cledumas, ABDULLAH Musa, YUSRI BIN KAMIN, RABIU HARUNA, and SHUAIBU HALIRU. "THE THE APPLICATION OF GENERIC GREEN SKILLS IN TESSELLATION OF REGULAR POLYGONS FOR ECONOMIC AND SOCIAL SUSTAINABILITY." Asia Proceedings of Social Sciences 4, no. 3 (June 16, 2019): 121–24. http://dx.doi.org/10.31580/apss.v4i3.873.
Full textAbstract:
Abstract
This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can free be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.
17
Nagel, Werner, and Viola Weiss. "Limits of sequences of stationary planar tessellations." Advances in Applied Probability 35, no. 1 (March 2003): 123–38. http://dx.doi.org/10.1239/aap/1046366102.
Full textAbstract:
In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.
18
Nagel, Werner, and Viola Weiss. "Limits of sequences of stationary planar tessellations." Advances in Applied Probability 35, no. 01 (March 2003): 123–38. http://dx.doi.org/10.1017/s0001867800012118.
Full textAbstract:
In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.
19
Gilevich, Artem, Shraga Shoval, Michael Nosonovsky, Mark Frenkel, and Edward Bormashenko. "Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs." Mathematics 12, no. 15 (August 5, 2024): 2426. http://dx.doi.org/10.3390/math12152426.
Full textAbstract:
A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as Rtrans(3,3)=5 Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ=limN→∞NgNr, where N is the total number of green and red seeds, Ng and Nr, were found ζ= 0.272 ± 0.001 (Voronoi) and ζ= 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.
20
Weydemann, Leonard, Christian Clemenz, and Clemens Preisinger. "On the Structural Properties of Voronoi Diagrams." KoG, no. 25 (2021): 72–77. http://dx.doi.org/10.31896/k.25.8.
Full textAbstract:
A Voronoi diagram is a tessellation technique, which subdivides space into regions in proximity to a given set of objects called seeds. Patterns emerging naturally in biological processes (for example, in cell tissue) can be modelled in a biomimicry process via Voronoi diagrams. As they originate in nature, we investigate the physical properties of such patterns to determine whether they are optimal given the constraints imposed by surrounding geometry and natural forces. This paper describes under what circumstances the Voronoi tessellation has optimal (structural) properties by surveying recent studies that apply this tessellation technique across different scales. To investigate the properties of random and optimized Voronoi tessellations in comparison to a regular tessellation method, we additionally run and evaluate a simulation in Karamba3D, a parametric structural engineering tool for Rhinoceros3D. The novelty of this research lies in presenting a simple and straightforward simulation of Voronoi diagrams and highlighting how and where their advantages over regular tessellations can be exploited by surveying more advanced approaches as found in literature.
21
Zhao, Yan, Yuki Endo, Yoshihiro Kanamori, and Jun Mitani. "Approximating 3D surfaces using generalized waterbomb tessellations." Journal of Computational Design and Engineering 5, no. 4 (January 9, 2018): 442–48. http://dx.doi.org/10.1016/j.jcde.2018.01.002.
Full textAbstract:
Abstract Origami has received much attention in geometry, mathematics, and engineering due to its potential to construct 3D developable shapes from designed crease patterns on a flat sheet. Waterbomb tessellation, which is one type of traditional origami consisting of a set of waterbomb bases, has been used to create geometrically appealing 3D shapes and been widely studied. In this paper, we propose a method for approximating target surfaces, which are parametric surfaces of varying or constant curvatures, using generalized waterbomb tessellations. First, we generate a base mesh by tiling the target surface using waterbomb bases. Then, by applying a simple numerical optimization algorithm to the base mesh, we achieve a developable waterbomb tessellation, which can be developed onto a plane without stretching. We provide a prototype system using which the user can adjust the resolution of the tessellation and modify waterbomb bases. Our work could expand the exploration of building developable 3D structures using origami. Highlights Generalizing waterbomb tessellations to fit target 3D parametric surfaces. Achieving developable tessellations by a simple numerical optimization algorithm. Non-axisymmetric or non-orientable resulting approximations are demonstrated.
22
Cowan, Richard, and Viola Weiss. "Line segments which are unions of tessellation edges." Image Analysis & Stereology 37, no. 1 (April 12, 2018): 83. http://dx.doi.org/10.5566/ias.1621.
Full textAbstract:
Planar tessellation structures occur in material science, geology (in rock formations), physics (of foams, for example), biology (especially in epithelial studies) and in other sciences. Their mathematical and statistical study has many aspects to consider. In this paper, line-segments which are either a tessellation edge or a finite union of edges are studied. Our focus is on a sub-class of such line-segments – those we call M-segments – that are not contained in a longer union of edges. These encompass the so-called I-segments that have arisen in many recent tessellation models. We study the expected numbers of edges and cell-sides contained in these M-segments, and the prevalence of these entities. Many examples and figures, including some based on tessellation nesting and superposition, illustrate the theory. M-segments are much more prevalent when a tessellation is not side-to-side, so our paper has theoretical connections with the recent IAS paper by Cowan and Thäle (2014); that paper characterised non side-to-side tessellations.
23
Erickson, Alyssa. "Exploring Geometry and Art through Tessellations." Journal of Technology-Integrated Lessons and Teaching 2, no. 2 (January 20, 2024): 56–58. http://dx.doi.org/10.13001/jtilt.v2i2.8389.
Full textAbstract:
In this lesson, elementary school students explore geometric shapes and tessellations using a Cricut Maker 3. During Part 1 of the activity, students review geometric concepts of regular versus irregular polygons and lines of symmetry. This includes using shapes cut by the Cricut machine to determine which regular polygons form a tessellation when put together. Then students answer reflection questions. During Part 2, there is a discussion about how the artist MC Escher used different types of symmetry (e.g., translations, rotations, and reflections) to modify irregular shapes to create tessellations. In Part 3, students are given materials to prototype their own tessellation using regular and irregular shapes and at least one type of symmetry transformation.
24
Brandts, Jan, Michal Křížek, and Lawrence Somer. "Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds." Symmetry 16, no. 2 (January 24, 2024): 141. http://dx.doi.org/10.3390/sym16020141.
Full textAbstract:
We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H2 by curved hyperbolic equilateral triangles whose vertex angles are 2π/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 H3. We also show that a regular tessellation of H3 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 H4. If n>4, then there exists no regular tessellation of Hn.
25
Clauss, Judith Enz. "Pentagonal Tessellations." Arithmetic Teacher 38, no. 5 (January 1991): 52–56. http://dx.doi.org/10.5951/at.38.5.0052.
Full textAbstract:
Investigating tessellations can furnish enjoyable and exciting mathe matics-inquiry activities for elementary and middle school students. The word tessellation come from a Latin word meaning “to pave with tiles,” or “to make a mosaic of.” When polygons are fitted together to cover a plane with no spaces between them and no overlapping, the pattern formed is called a tessellation. Creating and studying these patterns helps students to investigate basic properties of angles and polygons. Additionally, students develop skills in using rulers and protractors.
26
Maier, Roland, and Volker Schmidt. "Stationary iterated tessellations." Advances in Applied Probability 35, no. 2 (June 2003): 337–53. http://dx.doi.org/10.1239/aap/1051201649.
Full textAbstract:
The iteration of random tessellations in ℝd is considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.
27
Maier, Roland, and Volker Schmidt. "Stationary iterated tessellations." Advances in Applied Probability 35, no. 02 (June 2003): 337–53. http://dx.doi.org/10.1017/s000186780001226x.
Full textAbstract:
The iteration of random tessellations in ℝdis considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.
28
Giganti, Paul, and Mary Jo Cittadino. "The Art of Tessellation." Arithmetic Teacher 37, no. 7 (March 1990): 6–16. http://dx.doi.org/10.5951/at.37.7.0006.
Full textAbstract:
Students need to experience the excitement and beauty of mathematics beyond numerical calculations. Introduce your students to tessellations, a project that combines mathematics and art. A tessellation is a tiling, made up of the repeated use of polygons and other curved figures to completely fill a plane without gaps or overlapping, just like the tiles on a kitchen or bathroom floor. See figure 1.
29
Lachièze-Rey, R. "Mixing properties for STIT tessellations." Advances in Applied Probability 43, no. 01 (March 2011): 40–48. http://dx.doi.org/10.1017/s0001867800004675.
Full textAbstract:
The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations inRdwhich are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A∩M= ∅,ThB∩M= ∅) / P(A∩Y= ∅)P(B∩Y= ∅) − 1, whereAandBare both compact connected sets,his a vector ofRd,This the corresponding translation operator, andMis a STIT tessellation.
30
Ferraro, M., and L. Zaninetti. "On Non-Poissonian Voronoi Tessellations." Applied Physics Research 7, no. 6 (November 18, 2015): 108. http://dx.doi.org/10.5539/apr.v7n6p108.
Full textAbstract:
<p>The Voronoi tessellation is the partition of space for a given seeds pattern and the result of the partition depends completely on the type of given pattern ”random”, Poisson-Voronoi tessellations (PVT), or ”non-random”, Non Poisson-Voronoi tessellations. In this note we shall consider properties of Voronoi tessellations with centers gener-ated by Sobol quasi random sequences which produce a more ordered disposition of the centers with respect to the PVT case. A probability density function for volumes of these Sobol Voronoi tessellations (SVT) will be proposed and compared with results of numerical simulations. An application will be presented concerning the local struc-ture of gas (CO<sub>2</sub>) in the liquid-gas coexistence phase. Furthermore a probability distribution will be computed for the length of chords resulting from the intersections of random lines with a three-dimensional SVT. The agreement of the analytical formula with the results from a computer simulation will be also investigated. Finally a new type of Voronoi tessellation based on adjustable positions of seeds has been introduced which generalizes both PVT and SVT cases.</p>
31
Prvan, Marina, Julije Ožegović, and Arijana Burazin Mišura. "On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container." Mathematical Problems in Engineering 2019 (July 10, 2019): 1–16. http://dx.doi.org/10.1155/2019/9624751.
Full textAbstract:
In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean {3,122} semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE.
32
Lachièze-Rey, R. "Mixing properties for STIT tessellations." Advances in Applied Probability 43, no. 1 (March 2011): 40–48. http://dx.doi.org/10.1239/aap/1300198511.
Full textAbstract:
The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations in Rd which are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A ∩ M = ∅, ThB ∩ M = ∅) / P(A ∩ Y = ∅)P(B ∩ Y = ∅) − 1, where A and B are both compact connected sets, h is a vector of Rd, Th is the corresponding translation operator, and M is a STIT tessellation.
33
Abid, Awaludin, Kusrini Kusrini, and Amir Fatah Sofyan. "Pengaruh Konversi Nurbs Ke Polygonal Pada Desain Mobil 3d Terhadap Penilaian Kualitas 3d Model." Creative Information Technology Journal 6, no. 2 (February 24, 2021): 119. http://dx.doi.org/10.24076/citec.2019v6i2.250.
Full textAbstract:
Di Industri otomotif, biaya prototyping meningkat berbanding lurus dengan kompleksitas dan dependensi kendaraan. Sebagai alternatif untuk prototyping fisik dapat memanfaatkan teknologi baru seperti Augmented Reality (AR) dan Virtual Reality (VR) digunakan. Penggunaan VR dan AR melibatkan real-time rendering data CAD yang mengkonsumsi banyak memori dan mengurangi kinerja aplikasi. Persiapan data memiliki peran penting untuk meningkatkan kinerja sementara tetap mempertahankan topologi dan kualitas mesh. Proses optimalisasi data CAD yang digunakan yaitu Tessellation atau mengkonversi NURBS ke Polygons, berperan untuk menghasilkan output data yang memiliki efisien kinerja dengan topologi serta kualitas mesh yang baik. Hadirnya software 3D Data preparation dan optimasi pada kelas Tessellator. Autodesk Maya merupakan software pemodelan 3D yang mendukung Non-Uniform Rational Basis Spline ataupun CAD memiliki fitur mengkonversi model NURBS ke polygons, pemilihan kebutuhan atau requirement pada tessellation berpengaruh terhadap hasil output. Penilaian dilakukan menggunakan penilaian Objektif menggunakan 3D mesh visual quality metrics berbasis vertex-position Hausdorff Distance sehingga didapatkan requirement pada Tessellation yang efektif. Hasil dari konversi memiliki topologi yang serupa dengan software khusus data preparation dan optimasi, sedangkan hasil penilaian mesh visual quality metrics requirement yang mendekati yaitu menggunakan Tessellation Method Count dan General. Kata Kunci— Tessellation, Mesh Visual Quality, CAD, Polygon In automotive industry, cost of prototyping increases directly with complexity and dependencies of vehicle. As an alternative to physical prototyping can utilize new technologies such as Augmented Reality (AR) and Virtual Reality (VR) are used. And involves the real-time rendering of CAD data which consumes a lot of memory and reduces application performance. Data preparation has an important role to improve performance while maintaining topology and mesh quality. Process of optimizing CAD data used is Tessellation or converting NURBS to Polygons, whose role is to produce output data that has an efficient performance with topology and good mesh quality. Autodesk Maya is a 3D modeling software that supports Non-Uniform Rational Base Spline or CAD which has the feature of converting NURBS models to polygons, the selection of requirements or requirements on tessellation influences the output results. The assessment is done using objective assessment with 3D mesh visual quality metrics based on Hausdorff Distance vertex-position so that the requirements for effective Tessellation are obtained. The results of the conversion have a topology similar to special data preparation and optimization software, while the results of the mesh visual quality metrics requirement approach are close to using the Count and General Tessellation method. Keywords— Tessellation, Mesh Visual Quality, CAD, Polygon
34
Xu, Yong, Xiao Hong Fan, Ke Gao Liu, Lei Shi, Bin Xu, Fu Ming Wang, and Jun Pin Lin. "Applying and Practicing of MATLAB Programing for Voronoi Tessellation." Advanced Materials Research 706-708 (June 2013): 391–94. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.391.
Full textAbstract:
Three-dimensional (3D) Voronoi tessellation diagrams are generated by compiling program in MATLAB software. According to these 3D Voronoi cells, crystal structure, even the disorder structure, can be geometrically described and tessellated into the regular object, reflecting the atomic structural parameters, such as the coordination numbers (CN) and the atomic separation. Studying the characteristic of these tessellations, we can find the rule of atomic packing in real space. This Voronoi diagram is also useful in materials structure teaching curriculum and it is the significant basis to realized the atomic structure of the materials. By programing with MATLAB, the abstract concept of Voronoi tessellation are transformed into the lively and understandable computer animation, which can be easily recognized and mastered.
35
Phillips, Daisy. "Tessellation." Wiley Interdisciplinary Reviews: Computational Statistics 6, no. 3 (March 25, 2014): 202–9. http://dx.doi.org/10.1002/wics.1298.
Full text
36
Kendall, Wilfrid S., and Joseph Mecke. "The range of the mean-value quantities of planar tessellations." Journal of Applied Probability 24, no. 2 (June 1987): 411–21. http://dx.doi.org/10.2307/3214265.
Full textAbstract:
Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.
37
Kendall, Wilfrid S., and Joseph Mecke. "The range of the mean-value quantities of planar tessellations." Journal of Applied Probability 24, no. 02 (June 1987): 411–21. http://dx.doi.org/10.1017/s0021900200031053.
Full textAbstract:
Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.
38
Martínez, Servet, and Werner Nagel. "Regenerative processes for Poisson zero polytopes." Advances in Applied Probability 50, no. 4 (November 29, 2018): 1217–26. http://dx.doi.org/10.1017/apr.2018.57.
Full textAbstract:
Abstract Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.
39
Yamamoto, Yohei, Riku Nakazato, and Jun Mitani. "Method for solving origami tessellation hole problem using triangle twist folding." Journal of Computational Design and Engineering 9, no. 1 (February 2022): 144–54. http://dx.doi.org/10.1093/jcde/qwab074.
Full textAbstract:
Abstract Origami tessellations are geometric pieces folded from a single sheet of paper with flatly overlapped facets. Most existing origami tessellations are constructed by first marking a grid of crease lines on the paper and then arranging repeating patterns along the grid. However, this design method is limited because it cannot design origami tessellations with patterns that cannot be represented on a grid, such as a regular pentagon. This paper proposes a new construction method for origami tessellations that solves this problem and enriches these varieties. In the proposed method, a boundary of an origami tessellation is determined first, and then patterns called triangle twist fold patterns are placed inside the boundary. A similar approach is known as a hole problem, although in this paper, the problem is redefined and discussed in a form suitable for origami tessellations. As a result, a grid-independent construction method was proposed, and new origami tessellations were obtained by using software that implements the method.
40
Šedivý, Ondřej, Jules Mullen Dake, Carl Emil Krill III, Volker Schmidt, and Aleš Jäger. "DESCRIPTION OF THE 3D MORPHOLOGY OF GRAIN BOUNDARIES IN ALUMINUM ALLOYS USING TESSELLATION MODELS GENERATED BY ELLIPSOIDS." Image Analysis & Stereology 36, no. 1 (March 31, 2017): 5. http://dx.doi.org/10.5566/ias.1656.
Full textAbstract:
Parametric tessellation models are often used to approximate complex grain morphologies of polycrystalline microstructures. A big advantage of such models is the substantial reduction in disk space required to store large, three-dimensional data sets, especially when compared with voxel-based alternatives. By selection of an appropriate tessellation model, a reasonably small loss of information on the real grain shapes can usually be achieved. Special attention has recently been devoted to models based on ellipsoidal approximations fitted to each grain. Faces of these tessellations are portions of quadric surfaces whose parameters can be derived easily. In this paper, we deal with geometric features of the structure, notably curvatures and dihedral angles, which are closely related to the kinetics of grain growth. These characteristics are computed for ellipsoidbased tessellations fitted to two different aluminum alloys with nominal composition Al-3 wt% Mg-0.2 wt% Sc and Al-1 wt% Mg. The results are then compared with estimations based on meshed empirical data. We observe that the model offers more consistent estimations of grain shape characteristics than do the meshed empirical data. Precise description of grain boundaries by the model is also promising with respect to possible applications of these tessellations in stochastic space-time modeling of grain growth.
41
Mecke, J. "Inequalities for mixed stationary Poisson hyperplane tessellations." Advances in Applied Probability 30, no. 4 (December 1998): 921–28. http://dx.doi.org/10.1239/aap/1035228200.
Full textAbstract:
Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.
42
Mecke, J. "Inequalities for mixed stationary Poisson hyperplane tessellations." Advances in Applied Probability 30, no. 04 (December 1998): 921–28. http://dx.doi.org/10.1017/s0001867800008727.
Full textAbstract:
Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.
43
Chiu, S. N. "Limit theorems for the time of completion of Johnson-Mehl tessellations." Advances in Applied Probability 27, no. 4 (December 1995): 889–910. http://dx.doi.org/10.2307/1427927.
Full textAbstract:
Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝd and k-dimensional sectional tessellations, where 1 ≦ k < d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.
44
Chiu, S. N. "Limit theorems for the time of completion of Johnson-Mehl tessellations." Advances in Applied Probability 27, no. 04 (December 1995): 889–910. http://dx.doi.org/10.1017/s0001867800047728.
Full textAbstract:
Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝdandk-dimensional sectional tessellations, where 1 ≦k<d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.
45
Bejarano, Andres, and Christoph Hoffmann. "A generalized framework for designing topological interlocking configurations." International Journal of Architectural Computing 17, no. 1 (February 15, 2019): 53–73. http://dx.doi.org/10.1177/1478077119827187.
Full textAbstract:
A topological interlocking configuration is an arrangement of pieces shaped in such a way that the motion of any piece is blocked by its neighbors. A variety of interlocking configurations have been proposed for convex pieces that are arranged in a planar space. Published algorithms for creating a topological interlocking configuration start from a tessellation of the plane (e.g. squares colored as a checkerboard). For each square S of one color, a plane P through each edge E is considered, tilted by a given angle [Formula: see text] against the tessellated plane. This induces a face F supported by P and limited by other such planes nearby. Note that E is interior to the face. By adjacency, the squares of the other color have similarly delimiting faces. This algorithm generates a topological interlocking configuration of tetrahedra or antiprisms. When checked for correctness (i.e. for no overlap), it rests on the tessellation to be of squares. If the tessellation consists of rectangles, then the algorithm fails. If the tessellation is irregular, then the tilting angle is not uniform for each edge and must be determined, in the worst case, by trial and error. In this article, we propose a method for generating topological interlocking configurations in one single iteration over the tessellation or mesh using a height value and a center point type for each tile as parameters. The required angles are a function of the given height and selected center; therefore, angle choices are not required as an initial input. The configurations generated using our method are compared against the configurations generated using the angle-choice approach. The results show that the proposed method maintains the alignment of the pieces and preserves the co-planarity of the equatorial sections of the pieces. Furthermore, the proposed method opens a path of geometric analysis for topological interlocking configurations based on non-planar tessellations.
46
Thäle, Christoph, Viola Weiss, and Werner Nagel. "Spatial Stit Tessellations: Distributional Results for I-Segments." Advances in Applied Probability 44, no. 3 (September 2012): 635–54. http://dx.doi.org/10.1239/aap/1346955258.
Full textAbstract:
In this paper we consider three-dimensional random tessellations that are stable under iteration (STIT tessellations). STIT tessellations arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. In preparation for its proof, we obtain other distributional identities for the typical I-segment and the length-weighted typical I-segment, which provide new insight into the spatiotemporal construction process.
47
Thäle, Christoph, Viola Weiss, and Werner Nagel. "Spatial Stit Tessellations: Distributional Results for I-Segments." Advances in Applied Probability 44, no. 03 (September 2012): 635–54. http://dx.doi.org/10.1017/s0001867800005814.
Full textAbstract:
In this paper we consider three-dimensional random tessellations that are stable under iteration (STIT tessellations). STIT tessellations arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. In preparation for its proof, we obtain other distributional identities for the typical I-segment and the length-weighted typical I-segment, which provide new insight into the spatiotemporal construction process.
48
Weiss, Viola, and Richard Cowan. "Topological relationships in spatial tessellations." Advances in Applied Probability 43, no. 4 (December 2011): 963–84. http://dx.doi.org/10.1239/aap/1324045694.
Full textAbstract:
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.
49
Weiss, Viola, and Richard Cowan. "Topological relationships in spatial tessellations." Advances in Applied Probability 43, no. 04 (December 2011): 963–84. http://dx.doi.org/10.1017/s0001867800005255.
Full textAbstract:
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.
50
Soto-Johnson, Hortensia, and Dawn Bechthold. "Tessellating the Sphere with Regular Polygons." Mathematics Teacher 97, no. 3 (March 2004): 165–67. http://dx.doi.org/10.5951/mt.97.3.0165.
Full textAbstract:
Spherical geometry can be used to entice students into a deeper understanding of Euclidean geometry. Determining which regular spherical polygons tessellate the sphere is another motivating topic that is accessible to high school students. The most recognizable tessellations of the sphere are found on balls, such as soccer balls, volleyballs, and golf balls. Even Spaceship Earth at Epcot Center in Disney World involves tessellation.