Fractals in nature

A fractal fern

A fractal tree

How to grow a digital tree?

A fractal is an object with a fractional dimension!

0.6039

Other example of fractal: Koch’s snowflake D=log4/log3=1.261

Self-similarity in Koch’s curve

Two “classic” examples of fractal: the Julia set and the Mandelbrot set

How to create a Julia set? Consider the map f: z --> z^2 + c where z = x + iy = (x, y) and c = a + ib = (a, b) is a parameter in the mapping. It is equivalent to the two-dimensional map (Polar coordinate) r eiθ--> r^2 e2iθ+ c

This map of the complex numbers is equivalent to 3 successive transformations on the complex plane. Stretch points inside the unit circle towards the origin. Stretch points outside towards infinity Cut along the positive x-axis. Wrap the plane around itself once by doubling every angle. Shift the plane over so the origin lies on (a, b).

For other values of c we need a computer to find out the fixed points Despite all this stretching, twisting, and shifting there is always a set of points that transforms into itself. Such sets are called the Julia sets (after the French mathematician Gaston Julia who discovered them in the 1910s.) The Julia set for c = (0, 0) is easy to find: the set is the unit circle. For other values of c we need a computer to find out the fixed points

Examples of the Julia set on z plane

A Julia set is either totally connected or totally disconnected!

Self-similarity of the Julia set

An artistic visualization of the Julia set

Whether a Julia set is connected or not depends on the parameter c. Plot the Julia sets for all parameter values c. If the value of c makes the Julia set connected, then we say this c belongs to the Mandelbrot set. We can plot the Mandelbrot set on the c plane. (Note: the Julia set is defined on the z plane) Examine the Julia set to determine whether it is connected or not takes a long time. Luckily, we need to study only one point in the z plane: the origin If the origin never escapes to infinity then it is either a part of the Julia set or is trapped inside it. In both cases, the Julia set is connected. (Mandelbrot) (Note: If the origin is part of the set, the set is dendritic (branch-like). If it is trapped inside the set, the set is topologically equivalent to a circle.)

Mandelbrot set on the c plane (x,y)=(-2,0) (x,y)=(1/4,0) (x,y)=(-3/4,0) (x,y)=(0,0)

the bifurcation diagram! Mandelbrot set and the bifurcation diagram! 3 4 5 4 2 1 3 8

The first computer print-out of the Mandelbrot set All the ”islands” in the set are connected!!

The fascinating “universe” of the Mandelbrot set

The end

“Bulbs” with different periods

Period 3 3

Period 4 4

Period 5 5

Period 7 7

You can find thousands of artistic fractals on the web, for example...

etc...