Mathematicians can also imagine twisting that plane to produce another infinite shape called the helicoid.
For centuries, the plane and the helicoid were the only known examples of infinite, unbounded minimal surfaces that don't fold back to intersect themselves.
Although computer images and other evidence strongly suggested that the new surface met the criteria for placing it, alongside the helicoid and the plane, in the minimal-surface hall of fame, that wasn't enough for mathematicians.
15 Proceedings of the National Academy of Sciences, establishes that a particular shape--a helicoid with a handle--doesn't intersect itself.
Twisting the ordinary two-dimensional plane into a helicoid converts the plane's flatness into saddle-based curviness.
In the early 1990s, David Hoffman and Fusheng Wei, then at the University of Massachusetts at Amherst, and Hermann Karcher of the University of Bonn in Germany discovered complicated equations that seemed to represent a surface just like the helicoid but with a tunnel penetrating one of the levels.