Piecewise Isometries Plotter

December 28, 2024

Introduction

Piecewise isometries can generate intricate fractal patterns through simple rotation rules applied to different regions of the complex plane.

This post is about a particular family, defined by a piecewise function:

T(z) = \begin{cases} \alpha \cdot z & \text{if } |z - 2| > 1 \\ \alpha \cdot (\beta z + 2(1 - \beta)) & \text{if } |z - 2| \leq 1 \end{cases}

Where $\alpha$ and $\beta$ are complex numbers on the unit circle, determined by the angle factors.

Try adjusting the parameters below to explore different patterns:

α: 0.101

β: 0.641

Iterations: 1000

Initial Points: 250

Animation Steps: 1000

Interactive