`  :: The only reason I can see for separating the definition is to take some mystery out of it.  ``I know what regular polygons are, but what are uniform polygons?`` ``Well, they're all regular.`` ``Why didn't you say so?``  There's no inherent need to treat polygons as a special case: the regularity of uniform polygons follows from the same definition of uniform as applies to higher dimensions, combined with the limited repertoire of Flatland symmetries.  George's definition is not wrong – it produces an equivalent set – but it's unnecessarily complicated.  It is certainly worth mentioning that all uniform polygons are regular, but not as part of the definition of uniform.  :: Oops, hang on – there does need to be something about equal edges; a rectangle has a symmetry group transitive on the vertices.  —  `