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