Fractals with a Special Look at Sierpinski’s Triangle

By Carolyn Costello

What is a Fractal?

• Self-Similar• Recursive definition• Non-Integer

Dimension• Euclidean Geometry

can not explain• Fine structure of

arbitrarily small scale

Types of Fractals

• Iterated Function Systems

• Escape-Time

• Random

• Strange Attractor

Iterated Function System• Fixed geometric

replacement rule• Sierpinski’s Triangle (below)

by continuously removing the medial triangle

• Koch Curve (right) by continuously removing the middle 1/3 and replacing with two segments of equal length to the piece removed

Escape - Time

• Formula applied to each point in space.

• Mandelbrot Set start with two complex numbers, zn and c, then follow this formula, zn+1=zn +c and keeping it bounded

Random

• created by adding randomness through probability and statistical distributions.

• Brownian motion the random movement of particles suspended in a fluid (liquid or gas).

Strange Attractor

• start with some original point on a plane or in space, then calculate every next point using a formula and the

coordinates of the current point • Lorenzo’s attractor

use these three equations:

dx / dt = 10(y - x), dy / dt = 28x – y – xz, dz / dt = xy – 8/3 y.

What is the dimension? How do you know?

• Line

• Square

• Cube

Scale factor

Magnification Factor

Number of self-similar

Dimension

Line ½ 1

1/3 1

¼ 1

Square ½ 2

1/3 2

¼ 2

1/5 2

Cube ½ 3

1/3 3

¼ 3

1/53

What is the dimension? How do you know?

• Line

• Square

• Cube

Scale factor

Magnification Factor

Number of self-similar

Dimension

Line ½ 2 1

1/3 3 1

¼ 4 1

Square ½ 4 2

1/3 9 2

¼ 16 2

1/5 25 2

Cube ½ 8 3

1/3 27 3

¼ 64 3

1/5125 3

What is the dimension? How do you know?

Scale factor

Magnification Factor

Number of self-similar

Dimension

Line ½ 2 2 1

1/3 3 3 1

¼ 4 4 1

Square ½ 2 4 2

1/3 3 9 2

¼ 4 16 2

1/5 5 25 2

Cube ½ 2 8 3

1/3 3 27 3

¼ 4 64 3

1/55 125 3

• Line

• Square

• Cube

Dimension

• N= number of self- similar pieces• m = magnification factor• d = dimension

• N = md

• log N = log md

• log N = d log m

log N

D= log m

Dimension of the

Sierpinski Triangle

Log of the number of self-similar pieces

Dimension= Log of the magnification factor

Dimension of the

Sierpinski Triangle

= Log 3

Log 2

≈ 1.585

Log of the number of self-similar pieces

Dimension= Log of the magnification factor

Sierpinski’s Triangle

• Generated using a linear transformation• start at the origin

xn+1 = 0.5xn and yn+1=0.5yn xn+1 = 0.5xn + 0.5 and yn+1=0.5yn + 0.5

xn+1 = 0.5xn + 1 and yn+1=0.5yn

Sierpinski’s Triangle

Chaos Game

• The game starts with a triangle where each of the vertices are labeled differently, a die whose sides are marked with the labels of the vertices (two each) and a marker to be moved. Place the marker anywhere inside the triangle, then roll the die. Move the marker half the distance toward the vertex that appears on the die.

Sierpinski’s Triangle

• Pascal’s Triangle

Sierpinski’s Triangle

• Pascal’s Triangle mod 2

Sierpinski’s Triangle

• Pascal’s Triangle mod 3

Sierpinski’s Triangle

• Pascal’s Triangle mod 6