Cosmological symmetries

Abstract

The symmetries of polyhedra were primarily rotational. Turned around one of their axes of symmetry, they would overlap themselves. A body displaying n-fold symmetry around a given axis will, after n rotations, not just overlap itself in terms of the space taken up by its shape, but really return to exactly the same position: if we mark a particular point on it (e.g. the one we use to rotate it), it will return to the same place. If we continue to repeat the rotations of angle 2π/n, oneachnth occasion we will return to the same position (though, unless we place a mark on the polyhedron, we will not be able to distinguish the intermediate stages from one another). During the rotation, that is, the shape will periodically repeat itself.