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: