The sphere is a very special shape. For example, it is the only convex solid preserved by a three-dimensional symmetry group. It is conjectured to be a global minimum among convex solids of optimal packing volume fraction, and a local minimum of the random-close-packing volume fraction. I spend a lot of time thinking about how these properties and others vary as the sphere is deformed continuously into other shapes.
Here are 12 different ways to continuously deform the sphere. They have different symmetry properties: the first six all have an axis of continuous rotation symmetry. The next three groups of two have the tetrahedral group, the D3 dihedral group, and the octahedral group as their symmetry groups.
I calculated the rate at which the random-close-packing density of these shapes grow as one moves away from the sphere, and I made this javascript demo to make a figure for the paper I wrote on the calculation.