We describe a numerical algorithm to compute surfaces that are
approximately self-similar under mean curvature flow. The method
restricts computation to a two-dimensional subspace of the space of
embedded manifolds that is likely to contain a self-similar solution.
¶ Using the algorithm, we recover the self-similar torus of Angenent and
find several surfaces that appear to approximate previously unknown
self-similar surfaces. Two of them may prove to be counterexamples to
the conjecture of uniqueness of the weak solution for mean curvature flow
for surfaces.