A two-dimensional representation of a Klein bottle--a shape with no inside or outside, just one continuous surface. A true Klein bottle needs at least four dimensions; in other words, it can't be blown from glass. Two- and three-dimensional representations like this one exist to help us visualize the topology, but they are not completely faithful to the original shape. The surface cannot be built in two- or three-dimensional space without self-intersection, as shown here with the "handle" passing through the side of the surface.
Credit: Thomas Banchoff, Brown University, and Davide Cervone, Union College.
?. published on the web by University of Pittsburgh, School of Information Sciences, (2003)Report on the NSF Workshop on Research Directions for Digital Libraries, June 15-17, 2003, Chatham, MA.