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Putting a handle on a minimal helicoid.

The helicoid, resembling a spiral slide of infinite length, serves as an example of a minimal surface. Like a glistening soap film stretched across a wire frame, this figure's surface covers the least possible area within the helix defining its edge.

For centuries, the helicoid has been the only known example of a so-called "complete embedded minimal surface of finite topology with infinite total curvature." Roughly speaking, this means that the surface has no boundary and doesn't fold back to intersect itself. At the same time in the strange, rubbery world of topology, it's possible to imagine creating a helicoid by carefully deforming and stretching out the surface of a sphere punctured by a hole. The hole's rim becomes the helicoid's helical edge.

Now mathematicians have discovered a new minimal surface (pictured here) having the same basic properties as the helicoid. The novel surface incorporates what topologists call a handle. This feature, which looks like a hole in the fundamental helicoid shape, makes the new surface topologically equivalent to a punctured sphere equipped with a handle -- just like the one that sprouts from a mug.

The finding, made by David Hoffman and Fusheng Wei of the University of Massachusetts at Amherst and Hermann Karcher of the University of Bonn in Germany, is the latest in a series of discoveries that have greatly expanded the number of known examples of minimal surfaces of various types (SN: 3/16/85, p. 168; 9/3/88, p. 151). Hoffman described the new helicoid surface at a workshop on the visualization of geometric structures, held last week at the Mathematical Sciences Research Institute in Berkeley, Calif.
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Title Annotation:new minimal surface expands known number of minimal surfaces
Publication:Science News
Article Type:Brief Article
Date:Oct 24, 1992
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