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.
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 as+bs+cs+...=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.
- 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 -
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 -