On: Self-righting object - the Gomboc

The site says there are an infinite number of shapes with these properties. Anyone know if there are any that look dramatically different from theirs?

quantify different and I'm sure they can find it

How about scaling each shape to the unit volume, and then rotating to maximize the overlap. What's not overlapping is the shape delta.

With a different number of surfaces, for example?

Surface as defined by a non-differentiable border, sure. Topological differences could be easy to quantify. Extra points for knots and non-orientable surfaces (e.g. a klein bottle).