Help understanding Hausdorff dimension of Sierpinski triangle

Discussion in 'General Math' started by Johan Lans, Apr 22, 2007.

1. Johan LansGuest

I'm trying to understand this with fractal dimensions, and thought I
got it until I wanted to calculate the dimension of the Sierpinski
triangle.
The formula i use is d=log(N)/log(P/p) , where d is dimesion, N is
the number of units that fit into the larger object, P is the size of
the entire object, and p is the size of the smaller units.
Consider Sierpinski triangle made out of three smaller triangles, the
middle triangle is missing.
The size of P is 1 (one big triangle). The size of a small triangle
is 1/4 of the big one. N is three small triangles that fit into the
bigger one. This gives me d=log(3)/log(4), which is not the correct
answer. Where do i go wrong?

/Johan

Johan Lans, Apr 22, 2007

2. Robert IsraelGuest

"Size" here should refer to diameter, not area. The size of a
small triangle is 1/2 the size of the big one.

Robert Israel, Apr 22, 2007

3. Johan LansGuest

Robert Israel skrev:
Alright. thanks a lot. Is it the same way when dealing with three
dimensional objects?

/Johan

Johan Lans, Apr 22, 2007
4. israelGuest

It's the same in any metric space.

Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

israel, Apr 22, 2007