# Patterns in deep Mandelbrot zooms

A simple formula leads to emergent complexity.

# 1 Abstract

The Mandelbrot set emerges from iterations of \(z \to z^2 + c\). Zooming deep into the fractal, history repeats each time the navigator veers off-centre, doubled up and twice as fast. At each resulting crisis, decisions allow patterns to be sculpted in the intricate shape of the boundary. Patterns in the decisions made when zooming are reflected in the complex dynamics, in particular in the binary expansions of the pairs of external rays landing on the cusp of each baby Mandelbrot set copy at the centre of each phase. Numerical and symbolic algorithms are used to analyze the pattern of decisions given the final coordinates of artworks by several fractal artists. New families of patterns can be synthesized, and from them coordinates can be calculated for image generation. In this way the manual labour of constructing deep Mandelbrot zooms can be reduced, allowing more time for pattern design.

# 2 Publication

In open-access proceedings of Algorithmic Pattern Salon 2023.

Cite as:

Heiland-Allen, Claude. 2023. “Patterns in Deep Mandelbrot Zooms.” In Algorithmic Pattern Salon. Then Try This. https://alpaca.pubpub.org/pub/uqh84mma.

DOI coming soon.

# 3 Presentation

Paper session Friday 2023-11-24 14:00-15:00 UTC, online.

Watch the recording: Algorithmic Pattern Salon day 2.

# 4 Videos

These are the videos I prepared for the presentation:

# 4.1 Part 1

A zoom into Jonathan Leavitt’s Polefcra with narration and subtitles. 7m03s.

Made with FractInt converter for parameter conversion, Fraktaler-3 for fractal calculations, SubScript for narration and subtitling, zoomasm for zoom video assembly, and ffmpeg for encoding.

# 4.2 Part 2

An excerpt of a session making a tree (real time actual game play footage). No sound. 1m53s.

Before this video, about 40 seconds to find the starting embedded Julia set.

After this video, another 2 minutes continuing to period 77821.

Made with mandelbrot-perturbator for fractal exploration, OBS for screen recording, and mpv and ffmpeg for editing.