  In this work, we compute the perfect forms for all imaginary quadratic fields
of absolute discriminant up to $5000$ and study the number and types of the
polytopes that arise. We prove a bound on the combinatorial types of polytopes
that can arise regardless of discriminant and give a volumetric argument for a
lower bound on the number of perfect forms as well as a heuristic for a better
lower bound for imaginary quadratic fields of sufficiently large absolute
discriminant.
