A look inside the Mandelbrot set

The formula that generates the Mandlebrot set is this:

Z \rightarrow Z^2 + C

which can also be written:

Z_n = {Z_{n-1}}^2 + C

This formula is iterative: you get the value of Z at some iteration n (where n is an integer greater than zero) by squaring the value of Z at the previous iteration (iteration n-1) and adding C. When generating the Mandelbrot set, the initial value of Z (Z_0) is always 0; it is the value of C that is different for each point in the set.

Z and C are both complex numbers, numbers with both real and imaginary parts (e.g. x + yi, where i is the square root of -1). Now it happens that complex numbers, and arithmetic operations on complex numbers, can be translated into points in two different co-ordinate systems and operations on those points. If we can treat a complex number x + yi as a point in the cartesian co-ordinate system, \langle x, y \rangle, then we can treat the addition of two complex numbers as the geometric translation of one point by another:

Complex numbers: (x_1 + y_1i) + (x_2 + y_2i) = x_1 + x_2 + (y_1 + y_2)i
Cartesian points: \langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle = \langle x_1 + x_2, y_1 + y_2 \rangle

Additionally, if we convert our point in the Cartesian co-ordinate system into a point in a polar co-ordinate system \langle r, t \rangle , where {r} is the radius of the point (its distance from the origin) and t is the angle of the point in radians relative to the positive x axis, then we can treat the multiplication of two complex numbers as the geometric rotation and scaling of the point:

Complex numbers: (x_1 + y_1i) * (x_2 + y_2i) = x_1x_2 - y_1y_2 + (x_1y_2 + y_1x_2)i
Cartesian points: \langle x_1, y_1 \rangle * \langle x_2, y_2 \rangle = \langle x_1x_2 - y_1y_2,  x_1y_2 + y_1x_2 \rangle
Polar points: \langle r_1, t_1 \rangle * \langle r_2, t_2 \rangle = \langle r_1r_2, t_1 + t_2 \rangle

With this in mind, we can see that the formula Z \rightarrow Z^2 + C describes an operation analogous to that of taking a geometric point, squaring its distance from the origin and doubling its degree of rotation away from the positive x axis (call this “swirling” the point), and then translating it in a fixed direction by a fixed amount (call this “bodging” the point).

Now, the interesting thing about this combination of operations is that either of them taken by itself would result in a simple linear equation for Z_n. If the formula were Z_n = Z_{n-1} + C, then it could be rewritten simply as Z_n = Z_0 + (n * C) (in other words, just bodge Z_0 n times). If the formula were Z_n = {Z_{n-1}}^2, then this could be rewritten as Z_n = {Z_0}^{2^{n}} (swirl Z_0 n times). But by combining swirling and bodging (swirl-bodge-swirl-bodge, try it in Photoshop some time) we get a dynamic equation where the interactions between the swirls and bodges produce much more complex patterns.

The Mandelbrot set is the set of values of C for which the series of values of Z produced by repeated swirling and bodging converges on some point, and the points outside the set are those values of C for which the series of values of Z just go on getting bigger and bigger. There’s a shortcut we can use to tell if the latter is the case without knowing the entire (infinite) series: if the Cartesian point representing Z falls outside of a circle of radius 2 centred on the origin, then it’s fairly easy to prove that no subsequent value of Z will ever again fall inside that circle – points that “escape orbit” will diverge to infinity thereafter. So what computer programs that plot the Mandelbrot set tend to do is calculate the series up to a fixed number of iterations (say 256), and “give up” if the value of Z still hasn’t escaped orbit by then. What they’re really doing is plotting the points outside the Mandelbrot set (which are coloured according to the number of iterations it takes for them to escape orbit), and leaving uncoloured points which are not known to diverge. The Mandelbrot set thus appears in these graphs as a sort of black hole in the middle of a set of computed results, bordered by an undecided zone: points along the plotted perimeter might still eventually diverge if one iterated enough times, and as one increases the magnification and zooms in on a particular zone, the number of iterations needed to gain a distinct picture increases accordingly.

What sort of object is this set? It is, as the name implies, a multiple, separated from the set of complex numbers by an infinitely elaborate rule – infinitely elaborate, yet exhaustively describable in simple terms via recursion. It is not an object of human experience: what we experience, when we generate images of its exterior, is not the Mandelbrot set itself but a never-fully-complete subset of its complement. Does it exist? Or is it the spectral foreshadowing of a possible existent, posited by mathematics but nowhere completely realized?

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2 Responses to “A look inside the Mandelbrot set”

  1. son1 Says:

    “If the formula were Zn = Z(n-1)^2, then this could be rewritten as Zn = Z0 ^ n (swirl Z0 n times). But by combining swirling and bodging (swirl-bodge-swirl-bodge, try it in Photoshop some time) we get a dynamic equation where the interactions between the swirls and bodges produce much more complex patterns.”

    I think this is a typo. If Z_n = Z_{n-1}^2, then Z_n=(Z_0)^{2^{n}}.

  2. dominicfox Says:

    WordPress does embedded latex? Cool!

    Yes, you’re right, my mistake (and not a typo).


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