Fractals 2

We are used to shapes having dimensions 1, 2 or 3. Is it possible for a shape to have dimension not equal to an integer? What dimensions do fractals have?

First, we need to think of dimensions in a different way. Everyone should be happy with these statements:

  • If you make a ‘x k’ enlargement of a 1-dimensional shape, then k of the original shapes are required to make the new shape
  • If you make a ‘x k’ enlargement of a 2-dimensional shape, then k2 of the original shapes are required to make the new shape
  • If you make a ‘x k’ enlargement of a 3-dimensional shape, then k3 of the original shapes are required to make the new shape

Therefore, I could define the dimension of a shape as the power, d, to which k is raised in the statement:

  • when you make a ‘x k’ enlargement of the shape, then kd of the original shapes are required to make the new shape.

Now consider the ‘Pascal’s triangle’ fractal we mentioned last time. At each stage we need 3 of the next smaller versions down to make a new triangle which is a ‘x2’ enlargement.

Therefore, 2d = 3.

d is not an integer!!!

Figure / Dimension / No. of Copies
Line segment / 1 / 2 = 21
Square / 2 / 4 = 22
Cube / 3 / 8 = 23
Figure / Dimension / No. of Copies
Line Segment / 1 / 2 = 21
Square / 2 / 4 = 22
Cube / 3 / 8 = 23
Doubling Similarity / d / n = 2d
Figure / Dimension / No. of Copies
Line Segment / 1 / 2 = 21
Sierpinski's Triangle / ? / 3 = 2?
Square / 2 / 4 = 22
Cube / 3 / 8 = 23
Doubling Similarity / d / n = 2d