The Wente Torus
As concretized by Stewart Dickson Research sponsored by Computer Science and Mathematics Division of Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U.S. DOE under Contract No. DE-AC05-00OR22725

Is a compact immersion of the torus which has constant mean curvature. Previous to 1984, the only known surface having this property was the sphere. Henry Wente, who discovered this surface is on the faculty of the Department of Mathematics, University of Toledo.

Other images and discussion of the Wente Torus and other Constant Mean Curvature surfaces can be found at the following URLs:

The Wente Torus Page at The Mathematical Sciences Research Instutute. http://www.msri.org/publications/sgp/jim/images/stills/wente/mainc.htmlThe Wente Torus Page at the Geometry, Analysis, Numerics and Graphics (GANG) Center, University of Massachusetts, Amherst http://www.gang.umass.edu/gallery/cmc/cmcgallery0301.html

Here are some images I have made of the surface, from data supplied by Ivan Sterling.
See also: http://www.ifp.illinois.edu/~sdickson/Wente/Wente.html
https://web.archive.org/web/20070824155726/http://emsh.calarts.edu:80/~mathart/sw/objView/thicken.html