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Dynamical Systems of Simplices in Dimension Two or Three

  • Gérald Bourgeois
  • Sébastien Orange
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6301)

Abstract

Let T 0 = (A 0 B 0 C 0 D 0) be a tetrahedron, G 0 be its centroid and S be its circumsphere. Let (A 1,B 1,C 1,D 1) be the points where S intersects the lines (G 0 A 0,G 0 B 0,G 0 C 0,G 0 D 0) and T 1 be the tetrahedron (A 1 B 1 C 1 D 1). By iterating this construction, a discrete dynamical system of tetrahedra (T i ) is built. The even and odd subsequences of (T i ) converge to two isosceles tetrahedra with at least a geometric speed. Moreover, we give an explicit expression of the lengths of the edges of the limit. We study the similar problem where T 0 is a planar cyclic quadrilateral. Then (T i ) converges to a rectangle with at least geometric speed. Finally, we consider the case where T 0 is a triangle. Then the even and odd subsequences of (T i ) converge to two equilateral triangles with at least a quadratic speed. The proofs are largely algebraic and use Gröbner bases computations.

Keywords

Dynamical systems Gröbner basis Tetrahedron 

MSC

Primary 51F 13P10 37B 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Gérald Bourgeois
    • 1
  • Sébastien Orange
    • 2
  1. 1.FAA’A, Tahiti, Polynésie FrançaiseGAATI, Université de la polynésie française
  2. 2.LMAHUniversité du HavreLe HavreFrance

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