Fractals By Amanda Lewis What is a fractal? A fractal is defined - - PowerPoint PPT Presentation

Fractals

By Amanda Lewis

What is a fractal?

• A fractal is defined to be a rough or

fragmented shape that can be broken up into smaller parts, which can be seen as a smaller copy of the original shape.

Benoit Mandelbrot

• Born in Poland in

November 1924

• Known as the “father
• f fractal geometry”
• Coined the term

“fractal” in 1975

How long is the coast of Britain?

• Mandelbrot studied this question when he first

discovered the idea of fractals in nature.

• He concluded that in a sense, the coastline of

Britain is essentially infinite.

• By using smaller units of measurement, the

length of the coast increases.

Maps of Britain

Sierpinski Triangle

• Developed by

WacLaw Sierpinski

Nk = 3k

L k = (1/2)k = 2-k

Ak = L k

2 " Nk = (3/4)k

Let N be the number of triangles: Let L denote the length of sides of each triangle: Let A be the area of each triangle:

Koch curve

• The length of any line segment can be

described as being infinitely long.

• To develop the Koch Curve:
• Start out with a line segment.
• Divide into three segments.
• Replace middle segment with an

equilateral triangle.

Koch Snowflake

Nk = 4k " 3

Let define the number

• f sides after the kth step:

Nk

P

k = NkLk = 3(4 /3)k

P

k

Lk

Lk = 1 3k

Let define the length of each side after the kth step: Let define the perimeter of entire snowflake after the kth step:

Cantor Point Set

• Developed by Greg Cantor
• How to develop the Cantor Fractal:
• Start out with one line segment.
• Divide that segment into three different parts.
• Remove the middle third.
• What is left is two line segments and four

endpoints.

n 1/27 8 3 1/9 4 2 1/3 1 1 Length of line segments # of line segments Steps

2n

3"n

As we increase the amount of iterations, the length of the lines approaches zero:

Fractal Dimension

• Fractal dimension provides a way to measure

how rough fractal curves are.

D = logn log M

Where n = number of pieces M= the magnification factor (how many times the fractal has been magnified)

If the dimension is between 1 and 2, then it is a fractal

Dimension of the Koch Curve

• After doing the process just one time, there is one

line fragment that is divided up into four with an equilateral triangle in the middle. So, n = 4. Because these four pieces are 1/3 the length of the original line segment, we can say the magnification, M = 3.

D = log(4) log(3)

Because this number has a dimension greater than 1, then the Koch curve is a fractal.

= 1.26185...

Julia Set

• prisoner set: set of all complex numbers in

the functionʼs orbit that are bounded

• escape set: the complex numbers that are

unbounded in an orbit under a certain function.

Julia Set

• An example of a prisoner is

under the function,

z0 = 2

f (z) = z2 " z +1

z0 = 1 + i

z1 = f(1+i) = (1+i)2 - (1+i) +1 = i

z2 = f(i) = i2 - i +1 = -i

If we keep iterating, our values will switch back and forth between I and -I, so we call a prisoner. z0 = 2

Julia Set

• The Julia set is defined to be the boundary between

the prisoner set and the escape set, under the function:

f(z) = z2 + c

Where c represents a complex constant.

How are Fractals used in the real world?

• Fractals have been observed in just about every

living thing in nature from trees in the rain forest to

• ur human bodies.
• the Koch snowflake was used to make the antennas
• f our cell phones smaller while increasing the

amount of frequencies they can receive.

• Fractals are also used intensively in movies and

video games…

Star Wars: Episode III

• The idea of the fractal was taken and applied to a cylinder spiral

shape of lava. They took the original shape, shrunk it down and reapplied it. They repeated this over and over again to get a extremely realistic huge ball of fire and lava

The End