FRACTAL DIMENSION

DIMENSION

Point 0Line 1Plane 2Space 3

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

SIERPINSKI TRIANGLE IS SELF-SIMILAR

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.

Doubling the sides gives us three copies, so 3 = 2d, where d = the dimension.

SIERPINSKI TRIANGLE 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.