Probably the most famous and mathematically interesting Fractal is the Mandelbrot Set or M-Set. The M-Set is the set of complex numbers, , where the function () = 2 + does not diverge when iterated from z=0. An example of this for c=0, and c=1:
𝒄 =
𝒄 =
0 2 +0 = 0
o 2 +1= 1
0 2 +0 = 0
1 2 +1=2
0 2 +0 = 0
2 2 +1= 5
0 2 +0 = 0
5 2 +1= 26
When = 0 the function does not diverge. Whereas when = 1 the function clearly starts to diverge. This set has an infinitely complex border when plotted on the complex plain [fig 9]. This is due to its multifractal nature.
fig. 9: The M-Set plotted on a complex plain
One of the most interesting fallouts of the complexity of the boundary of the M-set is that, if you have a complex number which is on the boundary of the M-set, if you change a little bit, there is no way to predict what will happen, i.e. whether the function will diverge or not without calculating the function to thousands of iterations. There is an interesting relationship between the M-set and that of the Filled Julia Set. The Filled Julia Set of a function () = 2 + is defined as the set of all points, , for a given complex number, , that do not diverge. This means that in the Filled Julia Set, instead of always starting at z=0 as with
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