Introduction

What is an fractal ?

In the 19th century, amazing mathematical objects appeared own to Cantor : Fractals Curves. First, they were studied only for their beauty. But nowdays, scientists use them in order to study some physical phenomena. In the middle of the 20th century, the mathematician Benoit Mandelbrot was the first person to expose a theory about them and gave the name of Fractal.

Fractals can be divided in 3 groups :
- natural object like snow flake and fern
- geometrical figures
- mathematical theory

Fractals are mathematical functions whose equations are often simple but whose representations are always intricate and unforeseeable.

The King's Dream fractal

The King's Dream is a simple, yet beautiful fractal. The formula to produce it was developped by Clifford Pickover and published in his book Chaos in Wonderland.

To create this fractal, 2 equations, representing the X and the Y coordinates of the screen, must be iterated. The formula are for this fractal :

Xn = sin (Yn-1 * B) + C * sin (Xn-1 * B)
Yn = sin (Yn-1 * A) + D * sin (Yn-1 * A)

A, B, C and D are constants.

What was the study I made ?

This fractal is beautiful but very interessant too. I chose differing severals parameters in order to visualize different graphicals output.

In the first part, I made a computation with the data I found on the internet ( cf Bibliography ).

In a second part, severals constants and parameters were modified in order to study the little modifications.