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__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:

- 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!

- an infinite perimeter in a finite one -

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