FractalsBy Amanda Lewis

What is a fractal?• A fractal is defined to be a rough or

fragmented shape that can be broken upinto smaller parts, which can be seen asa smaller copy of the original shape.

Benoit Mandelbrot

• Born in Poland inNovember 1924

• Known as the “fatherof fractal geometry”

• Coined the term“fractal” in 1975

How long is the coast of Britain?

• Mandelbrot studied this question when he firstdiscovered the idea of fractals in nature.

• He concluded that in a sense, the coastline ofBritain is essentially infinite.

• By using smaller units of measurement, thelength of the coast increases.

Maps of Britain

Sierpinski Triangle

• Developed byWacLaw Sierpinski

!

Nk = 3k

!

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

!

Ak = Lk2 "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 bedescribed 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 = 4 k " 3

Let define the number of sides after the kth step:

!

Nk

!

Pk = NkLk = 3(4 /3)k

!

Pk!

Lk

!

Lk =13k

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

1/2783

1/942

1/311

Length of linesegments

# of linesegments

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 measurehow rough fractal curves are.

!

D =lognlogM

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 oneline fragment that is divided up into four with anequilateral triangle in the middle. So, n = 4. Becausethese four pieces are 1/3 the length of the originalline 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 inthe functionʼs orbit that are bounded

• escape set: the complex numbers that areunbounded in an orbit under a certainfunction.

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 betweenthe prisoner set and the escape set, under thefunction:

!

f(z) = z2 + c Where c represents a complex constant.

How are Fractals used in thereal world?

• Fractals have been observed in just about everyliving thing in nature from trees in the rain forest toour human bodies.

• the Koch snowflake was used to make the antennasof our cell phones smaller while increasing theamount of frequencies they can receive.

• Fractals are also used intensively in movies andvideo games…

Star Wars: Episode III

• The idea of the fractal was taken and applied to a cylinder spiralshape of lava. They took the original shape, shrunk it down andreapplied it. They repeated this over and over again to get aextremely realistic huge ball of fire and lava

The End