  Bishop's volume comparison theorem states that a compact $n$-manifold with
Ricci curvature larger than the standard $n$-sphere has less volume. While the
traditional proof uses geodesic balls, we present another proof using
isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area
for a given volume. Curiously, isoperimetric hypersurfaces can have codimension
7 singularities, an interesting challenge we are forced to overcome.
