Coasting along the fractal boundaries

I’ve been tootling around the fractal boundaries of the Mandelbrot Set with the XaoS viewer recently.

Here’s a few of the sights i’ve come across:

To think that such ‘organic’ looking complex, infinite, fractal, similar across scale, ‘as above so below’ forms can naturally arise from a simple rule repeated and fed back to itself!

Boundary conditions are strange and counter-intuitive. I continue in the struggle to understand what, if any significance this has. It certainly feels out of this world.

Read on for a brief ‘explanation’ of the Mandelbrot set.


Take a 2 dimensional plane with an x and y axis and 0 at the centre. Any coordinate on that plane would be what mathematicians call a complex number (made up of one of the regular numbers along the x axis, together with a number along the y axis, which mathematicians call ‘imaginary numbers’)

Any of these complex numbers (coordinates) can be put through a simple mathematical rule that starts with a certain value, makes a set amendment to that value and then feeds the result back into itself (The number of times this rule is carried out on a particular set of coordinates is termed its number of iterations)

(z) = z squared + c

Here, c denotes the coordinate on the 2 dimensional plane. A coordinate is deemed part of ‘The Mandelbrot set’ if, starting with z=0, the value of z does not exceed 2 after successive iterations of the rule (repetitions of the feedback)

The z value for coordinates outside the ‘Mandlebrot set’, do not remain bound after successive iterations and tend towards infinity.

It was only with the advent of the computer age that enough coordinates and iterations could be computed to present some sort of image on a 2 dimensional plane.

Mandelbrot set by Matelski and Brook (1980)

Here we see the coordinates that stay bound to a value not exceeding 2 after a number of iterations denoted by an asterisk.

Mandelbrot set detail

Now, with our unprecedented computer power we can compute each coordinate to thousands of iterations. Using colour gives more information. If a sequence for a particular coordinate escapes to infinity, We can colour it a different shade depending on how many iterations it takes to have a value greater than 2.

With this increase in computational power we discover that the real area of interest is not the shape of the bound set or the surrounding space running off towards infinity, but the boundary area where these areas interact. Counter-intuitively we find an infinite fractal realm: Chaos, but with a strange, yet familiar looking order. The only limit into how far we can zoom into this fractal boundary is dictated by how powerful our computers are.

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