Mandelbrot UnCurled
The Mandelbrot Set - when viewed it its entirety, simply as a black shape against a white background - is not the most beautiful thing in the world. It has been described as a squashed bug, a turtle, and probably many other un-graceful things (sorry turtles). In fact, Dr. Mandelbrot and his assistants apparently first mistook it for a splotch of ink on a test plot because they were using poor printers at the time of its discovery. But as we Mandelbrot-lovers know, what lies deep in the remote, microscopic crevices of the Set is simply intoxicating - and this is partly due to its popularity. But alas, that's not what this investigation is about - it is about the self-similarity of nodes (bumps) along the perimeter of the cartioid. The shape of the Mandelbrot Set has been explored by many mathematicians. Here are just a few of the many investigations:

http://classes.yale.edu/Fractals/MandelSet/MandelCombinatorics/N2Scaling/N2Scaling.html
http://linas.org/math/dedekind/dedekind.html

Mapping
The image above is the output of a Java program which plots the values of the Set "uncurled" (the two color images illustrate a mapping between polar coordinates and cartesian coordinates). The colors in the image are shown to indicate how this mapping is done. In the familiar cylindrical projections of the world's continents onto a globe, a line is pinched into a point when mapping the 2D image onto the sphere - specifically the bottom and top edges of the 2D image are mapped to the north and south poles. In the case of this mapping, the point at 0, 0 in the complex plane is mapped to the bottom edge of the uncurled image, and a circle centered at that point is mapped to the the top edge of the uncurled image.

Branching
Also, notice the number of branchings among consecutively-sized nodes, with the largest (green) one having one stem, while the next size down (the blue and yellow ones) each have two. The orange and purple ones have three, and so on. This branching scenario and other numerical propertires are quite involved and well-studied, as described in the book, The Beauty of Fractals. The red crevice, I assume, would contain one node which has an infinite number of branchings (and it would be infinitely small).

Next Version
I would like to make another version which uncurls everything outside of the main cartioid, traversing the perimeter of the cartioid instead of simply pivoting about the origin. In addition to this, I would like to explore what kind of scaling is necessary to make all the "main" nodes appear roughly the same size and shape. The goal is to essentialy factor out the things that are different, leaving similarities intact. The result would be a long row (actually, an infinite row - but I'll only show part of it) of nodes, all roughly the same size (though a simple uncurling would not result in perfect circles). Located in the spaces between these nodes would be smaller nodes, and in-between any pair of those would be even smaller nodes. And so-on. This can already be seen as you scan your eye around the cartioid. Will there be other patterns of similarity and difference? Probably!

This exploration, like many others using fractals, is initially motivated by aesthetics and visual curiosity, but there is an underlying mathematical beauty lurking beneath. The mathematical properties of the Set have been figured out to a large extent. But I am choosing to discover them on my own - because when I get there, I will arrive with a collection of images and interactive tools. And even if I never arrive at that mathematical understanding, my visual tools will be available for those of you who want to use them in your own journey, as you explore the deep mathematical beauty in the Mandelbrot Set.

Don't forget to check out MandelSwarm

If you have explored similar aspects of the Set (and if you have read this far, you apparently have some interest in this arcane topic!) please let me know. Email Jeffrey@Ventrella.com.