Dynamical Systems of Simplices in Dimension Two or Three

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.