The Mandelbrot set is created using this formula.

Z = Z x Z + C

Z = (0 + 0i)
C = (a + bi)

Multiply Z by itself. Add C. The answer is the new value for Z. Repeat until the absolute value of Z is greater than two.

So while it is probably one of the most complex objects in mathematics, it is created by iterating the simple formula above.

Z and C are complex numbers.

When it comes to complex numbers, eg: 1 + 2i. You can’t reduce them so that there’s only one term. You can’t derive a sum. You just have to write them down as 1 + 2i. These numbers, part real, part imaginary, are called complex numbers

The initial value of Z is zero, and the initial value of C is the location of the current pixel.

The complex plane is the two-dimensional space made up of real and imaginary numbers, where real numbers lie along the horizontal axis and the imaginary numbers lie along the vertical axis.
Complex plane
C is initialized to the complex number representing the point to calculate, its real portion is its horizontal distance from the center of the plane, its imaginary portion is its vertical distance from the center of the plane. So what happens? Will Z will start to become a very large number? or will it stay a very small number? – trapped around the center of the complex plane.

After the first iteration, Z is equal to C, because zero squared is zero. After the next iteration, if Z is larger than one, when it is squared, it will leap outwards, trying to break free. However, if C is located in the opposite direction, then when C is added in, it will pull Z back. If Z is smaller than one, squaring Z makes it even smaller. C is the wild card. where will it place Z? in or out?

There is only one way of knowing. Do the Calculations. Only since the early 1990’s have computers and fractal programs become advanced enough to make these kinds of fractal calculations practical. As you magnify the boundary of the Mandelbrot set, you can see that Z and C have had a long way to go before its decided if Z escapes out of the set or not. Constantly changing sides, always teetering near the brink of two, only to fall back towards zero. It is a battle where a fraction so small you can’t even comprehend it, is the difference between having an absolute value and staying forever trapped inside the set or shooting off into infinity.

click on “connectedness” to learn more about the mandelbrot set….