We'll describe calculus on two fractal sets: the unit interval and the
Vicsek set. Integration is pretty straightforward, but differentiation
is hard; we can only define a second derivative (a Laplacian).
Fortunately, in both cases we have spectral decimation, a powerful
algorithm which allows us to recursively construct both the eigenvalues
and eigenfunctions of the Laplacian. Along the way we'll relate all
this to resistor networks from high school physics, and maybe even learn
something new about sine and cosine.