Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

Defining noninteger dimensions

Mathematicians have studied objects with non-integer dimension since the early 20'th century. Pioneers were Russian and French Mathematicians. The topic became fashion in the 80'ies, when people started to generate fractals on computers. Evenso, like any fashion, the hype has faded, it is a beautiful corner of Mathematics. To define fractals, the notion of dimension is extended. Here is a definition of which illustrates this:
Define the s-volume of accuracy r of a set X as hs,r(X) = n rs, where n is the smallest number of cubes of side length r needed to cover X. The s-volume is then defined as the limit hs(X) of hs,r(X), when r goes to zero. The dimension is defined as the limiting value s, where hs(X) jumps from 0 to infinity.
A line segment X of length 1 in the plane can be covered with n intervals of length 1/n and hs,r(X) = n (1/ns). For s smaller than 1, this converges to 0, and diverges for s bigger than 1 for n going to infinity. The dimension is 1.
A square X of a plane of area 1 in space can be covered with n2 cubes of length 1/n and hs,r(X) = n2 (1/ns) which converges to 0 for s smaller than 2 and diverges for s bigger than 2. The dimension is 2.
The Shirpinski carpet is constructed recursively by dividing a square in the plane into 9 equal squares and cutting away the middle one, repeating this procedure with each of the remaining squares etc. At the k'th step, we need 8k squares of length 1/3k to cover the carpet. The s-volume hs,1/3k (X) of accuracy 1/3k is 8k (1/3k)s=8k/3ks, which goes to 0 for k approaching infinity if s is smaller than d=log(8)/log(3) and diverges for s bigger than d. The dimension of the carpet is d=log(8)/log(3)=1.893 a number between 1 and 2. It is a fractal.

 
Remark: the above given dimension is called "box counting dimension", a slightly simplified form of the Hausdorff dimension which is usually used.
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