Math E-320: Spring 2013
Teaching Math with a Historical Perspective
Mathematics E-320:Dimension
Instructor: Oliver Knill
Office: SciCtr 432


Since we missed the Analysis lecture due to global cancellations of classes on Monday, here is a small part of the presentation.
The main goal of this lecture is to be able to compute the dimension of fractals. The formula which little Spock learns as a teenager in the movie Startrek (2009) is
   dim  = - log(n)/log(r)
 
where n is the number of cubes of length r which are needed to cover the object. The dimension has to be understood in the limit when r goes to zero, but for all popular fractals, the above formula suffices. Anyway, since many objects in nature can be approximated by fractals like coast lines, plants, the lung or the brain or mountains, the number r is just taken to be a small number.

Lets look at the Sierpinski carpet. The first step of approximation is the following:


We have
    dim= -log(8)/log(1/3) 
because the side length of a square has length r=1/3 and there are 8 squares. Now, if we make the next approximation, we replace each square with this punctured square and get 8*8 = 64 squares and r=1/9.
But again
    dim= -log(64)/log(1/9) 
Of course, by the rules of the log, this is the same number. Its unfortunate that we could not look at the presentation together which consisted of many movies and animations. The following is just a tiny part:
Quicktime Movie:
Please send questions and comments to knill@math.harvard.edu
Math E320| Oliver Knill | Spring 2013 | Extension School | Harvard University