Fractals in nature. A fractal fern A fractal tree

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

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).

This map of the complex numbers is equivalent to 3 successive transformations on the complex plane.

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 t

hem 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 ofthe Julia seton z plane

A Julia set is either totally connected ortotally 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)=(1/4,0)

(x,y)=(-2,0)

(x,y)=(-3/4,0)

(x,y)=(0,0)

Mandelbrot set andthe bifurcation diagram!

12

4

8

3

45

3

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...