Journal ArticleDOI
Duality of isosceles tetrahedra
Jan Brandts,Michal Křížek +1 more
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In this article, the authors define a so-called dual simplex of an n-simplex and prove that the dual of each simplex contains its circumcenter, which means that it is well-centered.Abstract:
In this paper we define a so-called dual simplex of an n-simplex and prove that the dual of each simplex contains its circumcenter, which means that it is well-centered. For triangles and tetrahedra S we investigate when the dual of S, or the dual of the dual of S, is similar to S, respectively. This investigation encompasses the study of the iterative application of taking the dual. For triangles, this iteration converges to an equilateral triangle for any starting triangle. For tetrahedra we study the limit points of period two, which are known as isosceles or equifacetal tetrahedra.read more
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Morley’s trisector Theorem for isosceles tetrahedron
TL;DR: In this article, the authors extend Morley's trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space, and prove that the Morley tetrahedral of an IST is also an ISC.
Posted Content
Modified contact simplex iteration
TL;DR: In this paper, the root of the contact simplex is defined as a homothety image of the simplex with a special coefficient greater than 1, and the authors show that the circumcenter sequence of these simplices has two partial limits.
References
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Optimality of the Delaunay triangulation in źd
TL;DR: It is shown that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the DelaunayTriangulation, and the weighted sum of squares of the edge lengths is the smallest for Delaunays triangulations.
Book
Old and new unsolved problems in plane geometry and number theory
Victor Klee,Stan Wagon +1 more
TL;DR: In this article, Klee and Wagon discuss some of the unsolved problems in number theory and geometry, many of which can be understood by readers with a very modest mathematical background, and place each problem in its historical and mathematical context, and the discussion is at the level of undergraduate mathematics.
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On Nonobtuse Simplicial Partitions
TL;DR: In this paper, a survey of acute and nonobtuse simplices and associated spatial partitions is presented, including path-simplices, the generalization of right triangles to higher dimensions.
Journal ArticleDOI
Well-Centered Triangulation
TL;DR: This work presents an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed.
Journal ArticleDOI
Space-filling Tetrahedra in Euclidean Space
TL;DR: In the answer to the book-work question as discussed by the authors, one candidate stated that the three tetrahedra into which a triangular prism can be divided are congruent, instead of only equal in volume.