Which Tetrahedra Fill Space?

Senechal, Marjorie

doi:10.2307/2689983

Cited by 38 publications

(18 citation statements)

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“…To this end, we need the special tetrahedron P shown in denote the length of edge q¡q¡. The following two lemmas, which are proved in [10], are needed for Theorem 2 below. Lemma 1.…”

Section: Bisection Procedures Based On a Special Tetrahedronmentioning

confidence: 99%

On the Shape of Tetrahedra from Bisection

Liu¹,

Joe²

1994

Mathematics of Computation

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Abstract.We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes ¡Tü , ZTX , ST1, ... , which has the following properties: (1) Each mesh !Tn is conforming. (2) There are a finite number of classes of similar tetrahedra in all the ¿Tn , n > 0. (3) For any tetrahedron T? in ¿T" , n(T") > cxn(T), where n is a tetrahedron shape measure and cx is a constant. (4) <5(T?) < c2(l/2)"/3«5(T), where

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“…To this end, we need the special tetrahedron P shown in denote the length of edge q¡q¡. The following two lemmas, which are proved in [10], are needed for Theorem 2 below. Lemma 1.…”

Section: Bisection Procedures Based On a Special Tetrahedronmentioning

confidence: 99%

On the Shape of Tetrahedra from Bisection

Liu¹,

Joe²

1994

Mathematics of Computation

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“…In 3D it is not that easy, the regular tetrahedra do not tile 3D, see e.g. [8]. However, there have been shown many tilings of 3D so far.…”

Section: Introductionmentioning

confidence: 99%

Face-to-face partition of 3D space with identical well-centered tetrahedra

Hošek

2015

Appl Math

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“…The closer the triangle is to an equilateral triangle, the more accurately we can establish the coordinates of particular triangulation points by means of measurement of lengths of edges (and angles). Finally, nonobtuse simplices are also used in computer graphics [24], mathematical genetics [66], crystallography [32,77], in the finite difference method [49], and in the Monte Carlo method for solving partial differential equations [97, p. 210]. There are surely numerous applications and occurrences of nonobtuse and acute simplices that are not listed in this paper.…”

Section: Further Applicationsmentioning

confidence: 99%

“…. } define the corresponding Voronoi cells (also called Dirichlet regions [77]) in R d . Properties of these convex polytopes (see Figure 5) were studied by Voronoi (1868Voronoi ( -1908 in [92], though they were defined earlier by Dirichlet (1805-1859) as…”

Section: Acute Partitionsmentioning

confidence: 99%

On Nonobtuse Simplicial Partitions

Brandts¹,

Korotov²,

Křížek³

et al. 2009

SIAM Rev.

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On nonobtuse simplicial partitionsBrandts, J.H.; Korotov, S.; Kížek, M.; Šolc, J. Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Download date: 10 May 2018Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Dedicated to Ivan Hlaváček on his 75th birthdayAbstract. This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

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Which Tetrahedra Fill Space?

Cited by 38 publications

References 0 publications

On the Shape of Tetrahedra from Bisection

On the Shape of Tetrahedra from Bisection

Face-to-face partition of 3D space with identical well-centered tetrahedra

On Nonobtuse Simplicial Partitions

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