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MatheMUSEments
Weird Bottles
By Ivars Peterson
Muse, September 2001, p. 45.
An ordinary bottle has an inside and an outside. If an adventurous ant with
sticky feet were to walk along that bottle’s surface to get from the outside to
the inside, it would have to cross the lip that forms the bottle’s mouth.
Mathematicians have come up with a bizarre, mind-bending object they call a
Klein bottle that has no such lip, or edge. What appears to be the bottle’s
inside is smoothly connected with its outside!
To make a model of a Klein bottle, you could start with a long tube. You
would have to stretch and bend one of the tube’s ends so that it twisted around
and plunged through the tube’s side, then met the tube’s other end from the
inside.
People have tried doing this with glass, but it isn’t easy. Astronomer Cliff
Stoll of Oakland, California, first heard about Klein bottles when he was in
high school more than 30 years ago. He went to his chemistry lab, set up a Bunsen
burner, and tried to make one. “After burning my fingers and cracking a dozen
tubes, I gave up,” he says.
A few years later, while in college, Stoll tried again, this time with
gloves and better equipment. He failed once more. “Without enough heat, you can’t
bend the glass,” he explains. “Heat it too much and the glass melts into a
glob. And at the right temperature, it’s practically impossible to stretch
glass around a curve and down into itself.”
Stoll finally found the answer several years ago when he met some expert
glassblowers who had experience making intricate glassware for scientific
experiments. They showed him how to make a glass Klein bottle. Stoll crafted
his first bottle from a piece of lab glassware—a Pyrex 500-milliliter
Erlenmeyer flask with welded glass connections. He’s been making Klein bottles
ever since, even selling them to people who want an example of an object that
has no edges and only one surface.
When German mathematician Felix Klein discovered the mathematical object now
named for him, he probably had no idea that it would become an intriguing
challenge for glassblowers—and for anyone else who sees one of these weird
bottles and tries to figure out whether it will hold anything.
You can see examples of Cliff Stoll's Klein bottles at
http://www.kleinbottle.com/.
John Suillivan has computer-generated images of Klein bottles at
http://torus.math.uiuc.edu/jms/Images/klein.html.
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