In article <KWsuzBAEM44$EwF0@meden.demon.co.uk>,
Stewart Robert Hinsley <{$news$}@meden.demon.co.uk> wrote:
In article <d7ee230e.0312191421.59f19caa@posting.google.com>, fjkelly
fjkelly6112@hotmail.com> writes
can anybody explain to me the meaning of "dimension" of self-similar
objects (like fractals), how it's derived, and how this is used? for
instance, why is the "dimension" of the cantor set log2/log3?
If a self-similar figure is composed of copies all of a single size, and
which are disjoint sets, then the similarity dimension is
log(number of copies)/log(ratio of linear dimensions)
I.e. a line is composed of n line segments, and is n times the length of
each segment, and has dimension log(n)/log(n) = 1;
A square is composed of n^2 smaller squares, and its sides are n times
the length of the sides of each smaller square, and had dimension
log(n^2)/log(n) = 2.
This is the right idea. Try to cover your figure with disks of radius r
and ask how the number of disks needed changs with r. As the previous
poster noted, when you try to cover a square with these disks, you'll
need something like 1/r^2 of them, just by thinking about area.
That exponent "2" is there because the square is two-dimensional.
Do the same thing with a smooth curve: just by thinking about length,
you can see you'll need something like 1/r^1 disks of radius r just
to cover a curve (of length 1). I threw in an extraneous exponent of 1
there just to remind you that you are working with a 1-dimensional
thing when you speak of a curve. One more example: you need about
1/r^3 balls of radius r to cover some lumpy blob in 3-space; so
again we see the dimension of the blob (3) showing up as the exponent.
So here's a definition of dimension (of a compact subset X of R^n).
The dimension of X is the exponent d in this sentence:
When we cover X by balls of radius r, we need about 1/r^d of them.
More precisely, let N(r) be the minimal number of balls of radius r
whose union contains all of X ; then dim(X) = - log( N(r) )/log( r ) .
