Mandelbrot set

The Mandelbrot set is a subset of the complex line created from the set of all complex numbers

c

such that the sequence

|z_n|

where

z_0 = 0

and

z_{n+1}=z_n^2+c

is bounded.

You can find out if a number is or is not in the Mandelbrot set using this simple definition.

If you let c = 1, then with the formula, you can create the series 0, 1, 2, 5, 26 ..., which grows without bound, it diverges, meaning that 1 is not a member of the Mandelbrot set. If you let c = -1, then you create the series 0, -1, 0, -1, 0 .... Because the series does not diverge, -1 is a member of the Mandelbrot set.

Symbolically, the Mandelbrot set can be represented as so:

$M = \{c \in \mathbb {C}:(\exist s \in \mathbb {R}: \forall n \in \mathbb {N} (|P_c^n (0)| \leq s))\}$ where $P_c (z) = z^2 + c$ and $P_c^n (z)$ denotes the $n$ th iterate of $P_c (z)$

When plotted on the Argand plane, numbers that are in the Mandelbrot set are usually represented as black points on the plane, and ones that aren't are coloured, different colours representing different growth rates. The result is a very colourful, beautiful looking, and intricately designed image.

The set of all real numbers in the Mandelbrot set, the intersection of the Mandelbrot set and the real line $M \cap \mathbb {R}$ , is the closed interval [-2, 0.25], which means that every real number greater than or equal to -2 and less than or equal to 0.25 is in the Mandelbrot set. You can show that this is the case by plugging some element of the interval in the set for $c$ and creating a sequence.

The Mandelbrot set is a fractal which exhibits self-similarity, as shown when one zoom. It has a Hausdorff dimension, or fractal dimension of 2. The smaller regions inside the Mandelbrot set that exhibit similarity to the fractal are nicknamed "Minibrots". The largest and area of the Mandelbrot set is the large cardioid known as the main cardioid.