The Chaos Game Versus Uniform Rotation: From Sierpinski Gaskets to Periodic Orbits

TL;DR

This paper explores how different dynamical systems related to the Chaos Game generate fractals and periodic orbits, providing formulas for attractors and demonstrating how basins of attraction can produce novel tiling motifs.

Contribution

It introduces new dynamical systems related to the Chaos Game, derives formulas for their attractors, and shows how basins of attraction can create novel plane tilings.

Findings

Transition from random to uniform selection simplifies the Sierpinski gasket to periodic orbits.

Formulas for attractors depend only on contraction ratio and regular n-gon.

Basins of attraction can generate novel tiling motifs.

Abstract

In this paper, we introduce a couple of dynamical systems that are related to the Chaos Game. We begin by discussing different methods of generating the Sierpinski gasket. Then we show how the transition from random to uniform selection reduces the Sierpinski gasket to simple periodic orbits. Next, we provide a simple formula for the attractor of each of the introduced dynamical systems based only on the contraction ratio and the regular n-gon on which the game is played. Finally, we show how the basins of attraction of a particular dynamical system can generate some novel motifs that can tile the plane.