Whorls of Attraction

October 1, 2016

I recently purchased this most excellent ‘Kickstarter’ project, vintage style Mandelbrot map. Lovingly created by Bill Tavishttp://www.mandelmap.com.
He even went to the trouble of including a couple of the intrinsic attractor mappings…

mandelbrot_poster
Serendipitously, we were having a domestic Spring clean of accumulated detritus and I found this page among my personal effects. It dates from around 25 years ago. Ignoring the naïveté of the notation, screen co-ordinates & all, I thought I’d just leave this old print-out here for posterity, before it  gets trashed. It shows a graphic representation of the fractal reflection-translation used in my algorithm.
The bulbar cardoid of the Madelbrot may be famously familiar but the actual attractor-escape mappings are not so commonly illustrated, or much commented upon…

The closer an iterated point is to the central cartesian symmetry points x(-1,1) y(0), then characteristically, the stronger is the attractor’s ‘gravity’and the lesser the number of spiral limbs. Iterating points closer to the set’s escape boundary results in more complex spirals & increasingly chaotic ‘fingerprints’. From memory, the two centre shapes illustrated below, resulted from points on or just within the bulb’s boundary.

madelmap
So I’ve discarded one Mandelbrot picture only to frame another…
Haven’t fully decided what to do with the framed poster. If it looks too ‘school-roomish’ on my study wall, I may end up donating it to my kid’s school. Perhaps there, it might stimulate some errant student’s curiosity into discovering that mathematics holds deeper mystery and wonder than any dry school syllabus’ could ever convey?

 

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