You're mixing terms. A 71-sided polygon isn't a parallelogram, so to discard it because it doesn't have rotational symmetry is a straw man.

There are a set of properties that all polygons have (closed, etc), and a set of properties that all regular polygons have (regular angles, sides of the same length, and rotational symmetry at least 360/n degrees around the center, where n is the number of sides). This seems to lend itself to an inheritance tree, but, like the fibonacci sequence, the first approach isn't appropriate.

Instead, the more appropriate approach would seem to be to elaborate the rules that arise based on the properties of the object, not the class, and to (potentially) precalculate those results on object creation (with appropriate caching). For example, a square has certain properties. So, when an object of the ClosedShape class is instantiated that matches the properties of a square, the appropriate results would be calculated. But, it's still of the ClosedShape class.

I think that this would start to solve the generic and specific problems you have flyingmoose have raised.