Dynamical Systems of Simplices in Dimension 2 or 3

TL;DR

This paper studies a geometric dynamical system involving simplices in dimensions 2 and 3, proving convergence properties and explicit limits for the iterated constructions, with algebraic methods including Grobner basis computations.

Contribution

It introduces a new iterative process for simplices, proves convergence to regular shapes in 2D and 3D, and provides explicit formulas and convergence speeds, extending geometric understanding.

Findings

Sequences converge to equilateral triangles or isosceles tetrahedra.

Convergence speeds are at least quadratic or geometric.

Explicit edge lengths of limit shapes are derived.

Abstract

Let T_0=(A_0,..,A_d) be a d-simplex, G_0 its centroid, S its circumsphere, O the center of S. Let (B_i) be the points where S intersects the lines (G_0A_i), T_1 the d-simplex (B_1,..,B_d), and G_1 its centroid. By iterating this construction, a dynamical system of d-simplices (T_i) with centroids (G_i) is constructed. For d=2 or 3, we prove that the sequence (OG_i) is decreasing and tends to 0. We consider the sequences (T_{2i})_i and (T_{2i+1})_i ; for d=2 they converge to two equilateral triangles with at least quadratic speed ; for d=3 they converge to two isosceles tetrahedra with at least geometric speed. In this last case, we give an explicit expression of the lengths of the edges of the limit form. We show also that if T_0 is a planar cyclic quadrilateral then (T_n) converges to a rectangle with at least geometric speed or eventually to a square with a speed that is conjectured as cubic. The proofs are largely algebraic and use Grobner basis computations.