Two dynamical systems in the space of triangles

TL;DR

This paper studies two iterative maps on the space of triangles, showing convergence to equilateral and degenerate triangles, and discusses an analogous problem for quadrilaterals.

Contribution

It introduces and analyzes two specific dynamical systems on the space of triangles, revealing their convergence behaviors and extending the discussion to quadrilaterals.

Findings

Sequences under map f converge to equilateral triangle.

Sequences under map g converge to degenerate triangle with angles (0,0,π).

The maps demonstrate predictable geometric convergence behaviors.

Abstract

Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $\pi$ the triangle with angles numerically equal to edges of the initial triangle. The map $g$ corresponds to a triangle of perimeter $2\pi$ the triangle with \emph{exterior} angles numerically equal to edges of the initial triangle. For $p\in M$ the sequence $\{p,f(p),f(f(p)),\ldots\}$ converges to the equilateral triangle and the sequence $\{p,g(p),g(g(p)),\ldots\}$ converges to the "degenerate triangle" with angles $(0,0,\pi)$. In Supplement an analogous problem about inscribed-circumscribed quadrangles is discussed.