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, 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.

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 (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.

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.

Chaos | Quantum | Logic | Cosmos | Conscious | Belief | Elect. | Art | Chem. | Maths |

SCIENTIFIC AMERICAN March 1998
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