Translation SurfacesActually, the box isn't a translation surface. It is, however, in the more general class of flat surfaces. The difference between flat surfaces and translation surfaces has to do with cone points (see flat surfaces). The angle measure of a cone point is the sum of the angles of the polygons meeting at that point. If the sum is an integral multiple of {$2\pi$}, then the flat surface is a "translation surface." The torus and the Klein bottle can be expressed as translation surfaces. A 2-manifold is a translation surface iff the Euler characteristic is 0, in other words, the torus and the Klein bottle are the only 2-manifolds that are translation surfaces. |