5 - Fractal dimension and length considerations

a - Introduction to fractal dimension

How is defined the fractal dimension?
We begin to introduce the dimension for figure which dimension we know.

A line is one dimension object: to calculate this dimension, we write that when we divide a segment which length is one into two, we get two other identical segment which length equal one half.

__________ _____ _____
l=1 1/2 + 1/2

This can be written under the form: (1/2)^s+(1/2)^s=1 where s represent the dimension. As a matter of fact, 2x(1/2)^s=1 gives s=1.
In fact, the dimension is given by a relation a^s+b^s+c^s+...=1 where a, b, c, ... represent the size of every pattern appearing in the original one (which size is taken equal to one) and identical to it.

If we apply this for a square of 1x1. In this square, four squares appear, identical to the first one, with a side one half long. The previous relation is then:(1/2)^s+(1/2)^s+(1/2)^s+(1/2)^s=1.
very simply, we find that s=2: the dimension of an area.

For a cube, the number of identical patterns is eight, hence: 8x(1/2)^s=1 ==> s=3.

b - Dimension of the Koch curve

And what about our Koch's snowflake?

- decomposition -

This picture shows that the basis pattern is made of 4 other patterns which size are 1/3.
Consequently, the relations is:

What is the dimension?
- resolution of the dimension -

The dimension of the Koch curve is s=1,26185950714.

c - Considerations about the length

To draw the Koch curve, you begin with a line. Say its length is one. We can say that it is made of three segment with length 1/3. After one iteration, the figure is made of four such segments. So it length as grow to be 4/3. After another iteration, the length will be (4/3)x(4/3) or 16/9.
The length keeps increasing. After infinite iterations, the length of the Von Koch curve would be infinite.

Yet this infinite perimeter keeps staying into a finite one: the one of the circle that meets the three heights of the initial equilateral triangle!

Infinite perimeter into finite one!
- an infinite perimeter in a finite one -