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About the Mandelbrot set

1 March 2010

Image of the Mandelbrot set

The Mandelbrot set is the set of all complex numbers that satisfy a particular computational requirement. The equations are explained in the Mandelbrot Set Wikipedia article which also has some images. I have adopted the Mandelbrot set as my logo.

On the web there are many pages showing the beauty of the set as it is calculated with higher and higher resolution. I like the following images of the whole Mandelbrot set at larger scales:

2560 × 1920 image

3010 × 3030 image

10,240 × 10,240 image

With the ability to record slide shows, there are also Mandelbrot zooms, which give the impression of diving into the set as it calculated with increasing resolution. My favourite Mandelbrot set zoom video is on YouTube and can be watched below.

It shows how as you expand the set, the same images recur at smaller and smaller levels of details, which is one of the defining characteristics of fractals. No matter how much you expand the set, you never get a simple sharp edge; it is always fuzzy.

If listen to the soundtrack, it also explains the Mandelbrot set quite well. There are also references in the soundtrack to other fractals such as the Sierpinski gasket.

If you want to learn more about fractals, I recommend two books.

"Chaos: Making a New Science" by James Gleick

I read the first edition of this book in the late 1980s. It explains in a very readable way the difference between chaotic systems (which are theoretically predictable and governed by deterministic equations) and randomness. The weather is theoretically completely predictable, but small differences and measurement errors multiply up over time. Hence the saying about a butterfly flapping its wings in China leading to a typhoon elsewhere.

"The Fractal Geometry of Nature" by Benoit B. Mandelbrot

This is Mandelbrot's original masterpiece. It is very well written, but is quite a challenging book to read. I would not recommend it unless you have studied mathematics to at least UK GCE A Level, and preferably first year undergraduate level.

 

 

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