The Beauty of Fractals

Workshop Leaders

Stephanie Botsford

Sharon Bütz

Deb Carney

Allegra Reiber

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In a word or two ……

What does Mathematics mean to you?

Building a Koch Snowflake Fractal

1. Begin with an equilateral triangle.

2. Over the middle third of every line segment you see, erect an equilateral tent.

3. Erase the ground of the tent.

4. Return to Step 2 and repeat.

The Koch Snowflake

How would you describe a fractal?

What is a fractal?

Generally characterized as possessing two properties: Self-similarity Infinite Detail

Definition: A fractal is a set which (usually) has a non-integer dimension.

Let’s think about “non-integer dimension”.

Measuring the Koch Curve: Length

1st approximation of length: Distance from beginning point to end point

Total length: 1 unit

1 unit

Measuring the Koch Curve: Length

2nd approximation of length: Measure straight line distance with intermediate waypoints.

Total length: 4/3 = 1.3333... units

1/31/3

1/31/3

Measuring the Koch Curve: Length

3rd approximation of length: Measure straight line distance with more waypoints.

Total length: 16/9 = 1.7777... units

1/9

Measuring the Koch Curve: Length

Approximation Length

1 1

2 4/3 = 1.3333…

3 16/9 = 1.7777…

4 64/27 = 2.37037…

n

The Koch Curve has infinite length!

1

3

4lim

n

n

1

3

4

n

Measuring the Koch Curve: Area

A similar type of argument shows the Koch Curve has NO area.

What do “Infinite Length” and “No Area” suggest about the dimension of the Koch curve?

Dimension 1

Start with a line segment.A larger segment of double the size can be made in two ways.

Magnify by a factor of 2.

Arrange together 2 smaller segments.

This is because a line is 1-dimensional and 21 = 2.

Dimension 2 Object

Given a square, a larger square of triple the size can be made in two ways.

Magnify by a factor of 3. Arrange 9 smaller squares.

This is because a square is 2-dimensional and 32 = 9.

Dimension 3 Object

Given a cube, a larger cube of double the size can be made in two ways.

Magnify by a factor of 2. Arrange 8 smaller cubes.

This is because a cube is 3-dimensional and 23 = 8.

General Relationship

If a magnified object can be put together using copies of the object itself then the dimension is the number satisfying the equation

(magnification factor)dimension = number of copies

Dimension ?

The Koch Curve is d-dimensional where 3d = 4.

Magnify by a factor of 3

Assemble 4 copies

We can use logarithms to find d = 1.26186…

Fractal Activity!

Sierpiński Carpet: Count the number of white squares at each level.

Determine the total remaining area.

Guess what the next level will look like!

Fractal Activity: Area

Sierpiński Carpet:

What is a formula for the area at step 5?

How much area would there be after infinitely many steps?

Fractal Activity: Dimension

Let’s put our fractals together to make two large Sierpiński carpets.

How many more people would we need to make the next “level” of the fractal?

Fractal Activity: Dimension

(magnification factor)dimension = number of copies

What is the magnification factor?

How many copies?

3dimension = 8

Father of Fractal Geometry

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel a straight line.

- Benoit Mandelbrot

Benoit Mandelbrot: 1924 - 2010

Mandelbrot Set

Fractal Cookies

http://www.evilmadscientist.com/article.php/fractalcookies

Business Card Menger Sponge

Gwyneth and Ethan Ormes with a Level 2 Business Card Menger Sponge.

Business Card Menger Sponge

Dr. Jeannine Mosely inside her Level 3 Business Card Menger Sponge. http://theiff.org/oexhibits/paper06.html

Post-It Note Menger Sponge

Level 0

Level 2

(Each Post-it was torn into 16 pieces before folding into units.)

Level 3 (60% complete)

Real World Self-similarity

Real World Self-similarity

Romanesco Broccoli

Fractal Antennas

Dr. Nathan CohenFractal Antenna Systems Inchttp://fractenna.com

Fractals in Art

Jackson Pollock (1912 – 1956)

Hokusai (1760 – 1849)

Computer Graphics

Hunting the Hidden Dimension

For more on fractals there is an excellent NOVA Program: Hunting the Hidden Dimension. http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html