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Mathematical Recreations

by Ian Stewart

Glass Klein Bottles

Alan Bennett is a glassblower who lives in Bedford, England. A few years ago he became intrigued by the mysterious shapes that arise in topology - Möbius bands , Klein bottles and the like-and came across a curious puzzle. A mathematician would have tried to solve it by doing calculations. Bennett solved it in glass. His series of remarkable objects, in effect a research project frozen in glass, is soon to become a permanent exhibit at the Science Museum in London.

Topologists study properties that remain unchanged even when a shape is stretched, twisted or otherwise distorted the sole proviso being that the deformation must be continuous, so that the shape is not permanently torn or cut. (It is permissible to cut the shape temporarily, provided it is eventually reconnected so that points that were originally near one another across the cut end up near one another once again.) Topological properties include connectivity: Is the shape in one piece or several? Is the shape knotted or linked? Does it have holes in it?

The most familiar topological shapes appear at first sight to be little more than curious toys, but their implications run deep. There is the Möbius band, which you can make by taking a long strip of paper and gluing its ends together after giving the strip a twist. (Throughout this column, "twist" means "turn through 180 degrees," although sometimes this operation is known as a half-twist.) The Möbius band is the simplest surface that has only one side.

If two painters tried to paint a giant Möbius band red on one side and blue on the other, they would eventually run into each other. If you give the strip several twists, you get variations on the Möbius band. To a topologist, the important distinction is between an odd number of twists which leads to a one-sided surface, and an even number, which leads to a two- sided surface. All odd numbers of twists yield surfaces that, intrinsically, are topologically the same as a Möbius band. To see why, just cut the strip, unwind all twists save one and join the cut up again. Because you removed an even number of twists, the cut edges rejoin into a simple Möbius band.

Klein Bottle
KLEIN BOTTLE, a one-sided surface, blown in glass by Alan Bennett

For similar reasons, all bands made with an even number of twists are topologically the same as an ordinary cylinder, which has no twists. The precise number of twists also has topological significance, however, because it affects how the band sits in its surrounding space. There are two different questions here one about the intrinsic geometry of the band, the other about a band embedded in space. The first depends only on the parity (odd or even) of the number of twists; the second depends on the exact number.

The Möbius band has a boundary - those parts of the edge of the strip that don't get glued together. A sphere has no boundary. Can a one-sided surface have no edges at all? It turns out that the answer is yes, but no such surface can exist in three-dimensional space without crossing through itself.

This is no problem for topologists, who can imagine surfaces in space of higher dimensions or even in no surrounding space at all. For glassblowers, however, it is an unavoidable obstacle. The illustration above shows a glass Klein bottle blown by Bennett. Unlike an ordinary bottle, the "spout" or "neck" has been bent around, passed through the bottle's surface and joined to the main bottle from the inside. The glass Klein bottle meets itself in a small, circular curve; the topologist ignores that intersection when thinking about an ideal Klein bottle.

Imagine trying to paint a Klein bottle. You start on the "outside" of the large, bulbous part and work your way down the narrowing neck. When you cross the self-intersection, you have to pretend temporarily that it is not there, so you continue to follow the neck, which is now inside the bulb. As the neck opens up, to rejoin the bulb, you find that you are now painting the inside of the bulb! What appear to be the inside and outside of a Klein bottle connect together seamlessly: it is indeed one-sided.

Blown Glass Bottles
THREE-KNECKED Klein bottle( far left). NESTED SET of three Klein bottles (centre left). SPIRAL Klein bottle (center right) cuts into two seven-twist bands. VARIANT of spiral Klein bottle (far right).

Bennett had heard that if you cut a Klein bottle along a suitable curve, it falls apart into two Möbius bands. In fact, if you do this with a Klein bottle that sits in ordinary space like the glass one, those bands have a single twist. He wondered what kind of shape you had to cut up to get two three-twist Möbius bands. So he made many different shapes in glass and cut them up. He writes: "I find that if enough variations to the basic concept are made, or collected, the most logical or obvious solution to the problem usually becomes apparent."

Because he was looking for three-twist Möbius bands, Bennett tried all kinds of variations on the number three-such as bottles with three necks and, amazingly, sets of three bottles nested inside one another. He started to see, in his mind's eye, what would happen when they were cut up; he actually cut them up with a diamond saw to check.

Ouslam Vessel and Klein Bottle
OUSLAM VESSEL (right), whose neck loops around twice, separates into two three-twist Möbius bands if sliced vertically (centre). (The dotted lines are added as visual aids.) ORIGINAL Klein bottle (right) cut along a spiral curve.

The breakthrough was a very curious bottle whose neck looped around twice, forming three self-intersections. He named it the "Ouslam vessel," after a mythical bird that goes around in ever decreasing circles until it vanishes up its own rear end. If the Ouslam vessel is sliced vertically, through its plane of left-right symmetry-the plane of the paper in the drawing-then it falls apart into two three-twist Möbius bands. Problem solved.

Like any mathematician, Bennett was now after bigger game. What about five-twist bands? Nineteen-twist bands? What was the general principle? Adding an extra loop, he quickly saw that five-twist bands would result. Every extra loop put in two more twists.
Then he simplified the design, making it more robust, to produce spiral Klein bottles. The one depicted above, in the second photograph from the right, cuts into two seven-twist bands-and every spiral turn you add puts in two more twists. The photograph at the far right shows another variant on the same theme, a topological deformation of an ordinary spiral Klein bottle.

Mobius Bands
TWO MOBIUS BANDS result from cutting a Klein bottle along a curve.

Having now seen the significance of spiral turns, Bennett realized that he could go back to the original Klein bottle by "untwisting" the spiral. The line along which the spiral Klein bottle should be cut would deform, too. As the spiral neck of the bottle untwisted, the cut line twisted up. So if you cut an ordinary Klein bottle along a spiral curve, you can get as many twists as you want- in this case, nine.

Now for a final curiosity. The original motivation for the work was the possibility of cutting a Klein bottle to get two one-twist Möbius bands. But you can also cut a Klein bottle along a different curve to get just one Möbius band. I'll leave you to work out how, and I'll provide Bennett's solution in a future Feedback.





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