# Fractals

## Fractal Math

### Complex Numbers

The Mandelbrot set is a set of complex numbers, and a basic understanding of complex numbers is necessary to understand the definition of this set.

A complex number has the general format a+bi where a and b are real numbers while i is the imaginary unit which is defined as the square root of -1. A complex number thus has a real part a and an imaginary part bi. If z is a complex number such that z=a+bi, then a is sometimes referred to as Re(z) while b is referred to as Im(z) (Re - real part, Im - imaginary part).

Addition, subtraction, multiplication and division of complex numbers are defined as follows:

(a+bi) + (c+di) = (a+c) + (b+d)i

(a+bi) - (c+di) = (a-c) + (b-d)i

(a+bi) * (c+di) = (a*c-b*d) + (b*c+a*d)i

(a+bi) / (c+di) = (a*c + b*d) / (c*c + d*d) + ((b*c - a*d) / (c*c + d*d))i

The absolute value of a complex number is defined as:

| a+bi | = √(a2+b2)

The square of a complex number (according to the multiplication rule above) is:

a2-b2 + 2abi

### Complex Plane

A complex number can be viewed as a point in a two-dimensional Cartesian coordinate system, referred to as the complex plane. By convention, a is then the horizontal (x-axis) component, while b is the vertical (y-axis) component. The complex number 2.5+1.2i will then correspond to the point (2.5, 1.2) in the coordinate system.

### Mandelbrot Set

The Mandelbrot Set has the most well-known graphical visualization of mathematically derived fractal patterns. To understand the set, consider the complex quadratic polynomial

zn+1 = zn2 + c

where c is a complex parameter.

The following sequence can be defined from the polynomial above:

z0 = c

z1 = z02+c = c2+c

z2 = z12+c = (c2+c)2+c

:

For the complex parameter c = 1+2i the sequence develops as follows:

z0 = 1+2i

z1 = 12-22+2*1*2i+1+2i = -2+6i

z2 = (-2)2-62+2*(-2)*6i+1+2i = -31-22i

z3 = (-31)2-222+2*(-31)*(-22)i+1+2i = 478+1366i

:

It will easily been seen that this sequence will yield increasingly "big" numbers, that is, it is not bounded. A sequence which on the other hand is bounded, must be such that there exists a complex number s such that s is greater than zn for any n.

∀n∈{0,1,2,...}∃s∈R : |zn|  < s

For the complex parameter c = 0.5+0.6i the sequence develops as follows:

z0 = 0.5+0.6i

z1 = 0.52-0.62+2*0.5*0.6i +0.5+0.6i = -0.39+1.2i

z2 = (-0.39)2-1.22+2*(-0.39)*1.2i +0.5+0.6i = -0.7879-0.336i

:

This sequence look like it is bounded, however, there is generally no way to know this for sure (unless the same number appears for a second time in the sequence, i.e. there exists a j and a k where j ≠ k and zj = zk).

The Mandelbrot set is defined as the set of all complex numbers c for which the sequence zn+1 = zn2 + c  is bounded.