Transcript Fractal Dimension

FRACTAL DIMENSION
Point
Line
Plane
Space
0
1
2
3
DIMENSION
Similar - corresponding sides are in
proportion and corresponding angles
are of equal measure
Self Similar - each step is similar to each
other and to the original
Doubling of self similar
The length
The length and width
The length, width and height
DIMENSION IS THE EXPONENT
Start with a Sierpinski triangle of 1-inch sides. Double the
length of the sides. Now how many copies of the original
triangle do you have? The black triangles are holes, so we
can't count them.
SIERPINSKI TRIANGLE IS SELF-SIMILAR
Doubling the sides gives us three copies,
so 3 = 2d, where d = the dimension.
So the dimension of Sierpinski's Triangle is between 1 and 2.
Do you think you could find a better answer?
Use a calculator with an exponent key. Use 2 as a base and
experiment with different exponents between 1 and 2 to see
how close you can come.
For example, try 1.1. Type 2^1.1 and you get 2.143547. I'll bet
you can get closer to 3 than that. Try 2^1.2 and you get 2.2974.
That's closer to 3, but you can do better.
Okay, I got you started; now find the exponent that gets you
closest to 3, and that's its dimension.
SIERPINSKI TRIANGLE DIMENSION?