Details:
Title  Dynamical systems of simplices in dimension two or three.  Author(s)  Gerald Bourgeois, Orange Sébastien  Type  Book, Chapter in Book, Conference Proceeding  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  ISBN  9783642210457/pbk 
URL 
http://link.springer.com/chapter/10.1007%2F9783642210464_1 
Language  English  Pages  121  Publisher  Berlin: Springer  Year  2011  Edition  0  Translation 
No  Refereed 
No 
