Existence of an attractor for a geometric tetrahedron transformation

TL;DR

This paper investigates the dynamical behavior of a transformation on tetrahedra, proving the existence of a local attractor corresponding to regular tetrahedra and conditions for global attraction.

Contribution

It establishes the existence of a local attractor for the tetrahedron transformation and identifies conditions for global attraction in the space of tetrahedra.

Findings

Existence of a local attractor at regular tetrahedra

Conditions under which the entire space is attracted to the set of regular tetrahedra

Numerical evidence supporting the theoretical conditions

Abstract

We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric $8$-dimensional space $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. We establish the existence of a local attractor which coincides with the set of regular tetrahedra and identify conditions which imply that the basin of attraction is the entire space. In numerical tests, these conditions are fulfilled for a large set of random tetrahedra.