Fractals
Dimensions
Traditional Informal Definition:
The dimension of a set is the number of perpendicular direction
pairs you can move and stay within the set.
Modified Cantor Set
st
1 Generation
Modified Cantor Set
nd
2 Generation
Modified Cantor Set
rd
3 Generation
Modified Cantor Set
th
7 Generation
2-D or 1-D?
Fractals
Common (but too broad) Informal Definition:
A fractal is an object that displays some degree of self-similarity.
( The problem with this definition is that according to this definition a line segment is a
fractal since any part of it is also a line segment.)
Formal Definition from Benoît Mandelbrot :
A fractal is an object whose topological dimension is strictly less than the Hausdorff
dimension .
( The problem with this definition at the College Algebra level is that we don’t have enough
information to give precise definitions of the 2 dimensions, but I will give you some ideas
about these dimensions in the next slides.)
What I Think Is The Best Informal Definition For This Level:
A fractal is an object whose dimension is debatable.
Topological Dimension
Though the formal definition is beyond this level, basically the topological dimension of a
set is the same as the definitions we had at the beginning of the presentation.
0-D: made up of disconnected points.
1-D: made up of curves.
2-D: made up of surfaces.
3-D: made up of solids.
Thus, the topological dimension of any non-empty set is always a whole number
(0,1,2,3,…), and will never have any decimal part.
The topological dimension of the modified Cantor set is 1 since it is ultimately made up of
lines.
Hausdorff and other Fractal
Dimensions
Though again the formal definition is beyond this level, the Hausdorff and other fractal
dimensions note that the method of measuring a set is dependant of its dimension.
The number of points.
The length of a curve.
The area of a surface.
The volume of a solid.
Also, the formulas used depend on a exponent. For example, using some simple objects:
The number of a single point = 1 = s0
The length of a line segment = s = s1
The area of a square = s2
The volume of a cube = s3
Note that the exponent is also the dimension of the object.
Since the fractal dimension of a set is related to the exponent and exponents can have a
decimal part, a set’s fractal dimension can be any non-negative value, i.e. in the interval
[0,∞).
Another Fractal Dimension:
The Box-Counting Dimension
1-D Line:
Each time the
length of the sides
of the boxes is
divided by 2 it
takes 2 times as
many boxes.
Divider=2
Multiplier=2=21
Dimension=1
2-D Square:
Each time the
length of the sides
of the boxes is
divided by 2 it
takes 4 times as
many boxes.
Divider=2
Multiplier=4=22
Dimension=2
The Box-Counting Dimension on the
Modified Cantor set
Dividing the length of the sides of the
box(rectangle) by 3 it takes 6 times as
many boxes.
Divisor=3
Multiplier=6=3?
3d  6
 
ln 3d  ln  6 
d ln  3  ln  6 
d
ln  6 
ln  3
d  1.63
Thus, the modifies Cantor set has a
box-counting dimension of about
1.63. So it’s between a line and a
surface, and hence it’s a fractal.
More on Fractals
Wolfram-Mathworld Online Math Encyclopedia
http://mathworld.wolfram.com/Fractal.html
Zooming in on the Mandelbrot Set
http://www.youtube.com/watch?v=G_GBwuYuOOs
http://www.youtube.com/watch?v=vuyRCfhCZT0
NOVA Hunting the Hidden Dimension
http://www.pbs.org/wgbh/nova/fractals/
Some Applications of Fractals
Antennas
Image Compression
Computer Animation