Mandelbrot set love

# Mandelbrot set

## Definitions

### from The American Heritage® Dictionary of the English Language, 4th Edition

• n. The set of complex numbers C for which the iteration zn+1 = zn2 + C produces finite zn for all n when started at z0 = 0. The boundary of the Mandelbrot set is a fractal.

• n. the set of complex numbers c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. The boundary of this set is a fractal.

• n. a set of complex numbers that has a highly convoluted fractal boundary when plotted; the set of all points in the complex plane that are bounded under a certain mathematical iteration

## Etymologies

After Benoit B. Mandelbrot (born 1924), Polish-born American mathematician.
(American Heritage® Dictionary of the English Language, Fourth Edition)

• There is an extrememely amusing song by Jonathan Coulton, called Mandelbrot set. It is also a little helpful if you are not really a mathmetician.

Pathological monsters! cried the terrified mathematician
Every one of them is a splinter in my eye
I hate the Peano Space and the Koch Curve
I fear the Cantor Ternary Set
And the Sierpinski Gasket makes me want to cry
And a million miles away a butterfly flapped its wings
On a cold November day a man named Benoit Mandelbrot was born

His disdain for pure mathematics and his unique geometrical insights
Left him well equipped to face those demons down
He saw that infinite complexity could be described by simple rules
He used his giant brain to turn the game around
And he looked below the storm and saw a vision in his head
A bulbous pointy form
He picked his pencil up and he wrote his secret down

Take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set

Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire