Piecewise Isometries Plotter

December 28, 2024

α 0.101 β 0.641

drag on the image to steer · ←→ α ↑↓ β · hold shift for fine steps

parameter spaceα → · β ↓

α 0.101

β 0.641

iterations1000

points250

spin speed3

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About

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.