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 glasswarea 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 glassblowersand for anyone else who sees one of these weird bottles and tries to figure out whether it will hold anything.
Muse, September 2001, p. 45.
You can see examples of Cliff Stoll's Klein bottles at http://www.kleinbottle.com/.
John Sullivan has computer-generated images of Klein bottles at http://torus.math.uiuc.edu/jms/Images/klein.html.