This document discusses fractals and their properties. It provides examples of fractals like the Koch curve and Sierpinski carpet to illustrate self-similarity and fractional dimension. Workshop leaders describe how fractals appear in nature and how their non-integer dimensions can be calculated. Examples are given of fractals constructed from business cards and post-its. Fractals are also discussed in art and antenna design.

1. The Beauty of
Fractals
Workshop Leaders
Stephanie Botsford
Sharon Bütz
Deb Carney
Allegra Reiber

2. What does Mathematics mean to
you?

3. 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.

4. How would
you describe
a fractal?

5.  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”.

6. 1 unit
1st approximation of length: Distance from beginning point to
end point
Total length: 1 unit

7. 1/ 1/
3 3
1/ 1/
3 3
2nd approximation of length: Measure straight line distance with
intermediate waypoints.
Total length: 4/3 = 1.3333... units

8. 1/
9
3rd approximation of length: Measure straight line distance with
more waypoints.
Total length: 16/9 = 1.7777... units

9. Approximation Length
1 1
n 1
2 4/ = 1.3333… 4
3
lim 3
3 16/
9 = 1.7777… n
4 64/
27 = 2.37037…
n 1
n 4
3
The Koch Curve has infinite length!

10.  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?

11. 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.

12. 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.

13. 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.

14. 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

15. Magnify by a factor of 3
Assemble 4 copies
The Koch Curve is d-dimensional where 3d = 4.
We can use logarithms to find d = 1.26186…

16. Sierpiński Carpet:  Count the number of
white squares at each
level.
 Determine the total
remaining area.
 Guess what the next
level will look like!

17. Sierpiński Carpet:  What is a formula
for the area at step
5?
 How much area
would there be after
infinitely many
steps?

18.  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?

19. (magnification factor)dimension = number of copies
 What is the
magnification factor?
 How many copies?
 3dimension = 8


20.  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:
1924 - 2010
- Benoit Mandelbrot

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

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

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

25. (Each Post-it was torn into 16 pieces before folding into units.)
Level 0 Level 2
Level 3 (60% complete)

27. Romanesco Broccoli

28. Dr. Nathan Cohen
Fractal Antenna Systems Inc
http://fractenna.com

29. Jackson Pollock (1912 – 1956) Hokusai (1760 – 1849)

31.  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