If you follow this blog regularly, you know that I love fractals. I recently spent a pleasant afternoon in Nashua, New Hampshire, where my daughter Teresa introduced me to Gregory Searle, a fractal artist and computer geek. Here is his logo.

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Fractal Generator

Greg's web page is <http://www.fractalartdesign.com>. There is much to explore. To get started, click on "Fractal Generator" in the menu on the left of the page, and then on the "Fractal Generator" icon in the center of the next page. Or go directly to <http://fractal.fractalartdesign.com>.

You will find yourself in an app, written in JavaScript and running in your browser, where you can endlessly explore the Mandelbrot set and an unlimited collection of variations. You can zoom, pan, change iterators, vary parameters, and select from almost two dozen themes or color maps. You can edit the colormaps and investigate smoothers, renderers, and other effects.

This is the tool that Greg uses to produce his art. He writes, "A painter uses paints; I use a CPU. I am giving you access to my paints! Please be aware that this is always a work in progress. It will change at a whim."

There is a lot of fractal art on the Web. This tool allows you to create your own.

Double Double

Zooming in on a Mandelbrot or similar fractal by a factor approaching $2^{53}$ reaches the limit of IEEE double precision floating point. So, Greg's generator switches to double double precision. This doubles the fraction length and slows the rendering significantly. It does not increase the exponent, but these calculations need more precision, not more range. (See https://blogs.mathworks.com/cleve/2017/05/22/quadruple-precision-128-bit-floating-point-arithmetic.)

Aquifer

Greg uses his generator, and his skill, to create unique single-edition prints. To preserve their value, he does not reveal the parameters and never reprints. He displays his work at galleries and exhibitions around New England. Some of it is shown, and offered for sale, on his web site.

Here is a sample.

Aquifer

Amphibian

Amphibian

Clematis

Clematis

Wavelets

Here is one of his wallpapers, digital images that are available as a free download for use as a desktop background or device lock screen.

Wavelets

Mandelbar

Greg did not make the next two images -- I did, using his generator and nonstandard iterators. The color themes and shading are novel. This one is known on the Internet as the "Mandelbar" set because the iterator uses the complex conjugate, with a bar over the $z$,

$$z = \bar{z}^2 + c$$

instead of the traditional

$$z = z^2 + c$$

Mandelbar

Burning Ship

And this amazing fractal, known as the "Burning Ship", is produced by zooming in on an iterator involving the complex absolute value.

$$z = |z|^2 + c$$

Burning Ship

To see much more of the Burning Ship fractal, check out this video, https://www.youtube.com/watch?v=CD9yNFmb2FE.

Get the MATLAB code

Published with MATLAB® R2018a