September 25, 2007

Folding a Klein Bottle

The intriguing mathematical object known as a Klein bottle has only one surface. In effect, it has an outside, but no inside.

In abstract terms, you can make a Klein bottle by starting with a rectangular piece of the plane. You join two opposite edges to create a tube. You then join the remaining two edges, but with the same sort of twist (so that the joined edges have opposite orientation) that you need to make a Möbius strip. The second step can't be carried out without the tube passing through itself.


A true Klein bottle exists only in a topologist's imagination. It has to intersect itself without the presence of hole—a physical impossibility. Nonetheless, that hasn't stopped illustrators and glassblowers from attempting to represent this remarkable object in some physical form. Movies.

To make one from glass, for instance, you could start with a stretchable glass tube. One end of the tube narrows into a long neck; the other end widens into a base. The neck passes through the side of the bottle, and the neck and the base join to make the neck's inside continuous with the base's outside.

Astronomer Cliff Stoll is among those who have succeeded in crafting glass Klein bottles. He now offers them for sale, in a wide variety of sizes and manifestations, at Acme Klein Bottle.


A genuine Acme Klein Bottle. Photo by I. Peterson.

Several years, Stoll invited Robert J. Lang, creator of intricate, highly original origami constructions—insects, lobsters, birds, mammals, cuckoo clocks, and more—to construct a Klein bottle from a sheet of paper. It wasn't easy, but Lang met the challenge, folding a 13-inch-by-30 inch rectangle into a 7-inch-tall paper model.



Lang later made available the crease pattern that he used to create his paper Klein bottle. Lang displayed his Klein bottle and various other objects at MathFest in August when he appeared at the A K Peters booth to sign copies of his book Origami Design Secrets: Mathematical Methods for an Ancient Art.


There are other frontiers in the realm of Klein bottle visualization. The bottle form is just one of several ways to represent this mathematical object. Indeed, the Klein surface can take on many shapes that don't look at all like bottles. One version looks like a ring pinched so that it has a figure-eight cross section. But the orientation of the figure eight changes in going around the ring.


On the construction side, Andrew Lipson specializes in building mathematical objects out of LEGO bricks. He has made a hinged Klein bottle out of bricks and other pieces. His hinged construction shows how a Klein bottle, cut in half lengthwise, becomes two Möbius strips.

References:

Banchoff, T.F. 1998. Surfaces Beyond the Third Dimension: The Klein bottle. In Communications in Visual Mathematics.

Lang, R. 2003. Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters.

Peterson, I. 2001. Immersed in Klein bottles. MAA Online (Feb. 19).

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