Impossible Bottle Built

"All the news we feel like printing"

The Klarreich Occasionally

Vol. IV, No. 1

September 8, 2002

IMPOSSIBLE BOTTLE BUILT

In a feat mathematicians had dismissed as impossible, Herbert Gassner, a dentist from Paterson, New Jersey, has built the paradoxical bottle called a Klein bottle, which has no inside or outside. The physical manifestation of the bottle means that mathematicians must revisit their most fundamental assumptions about how mathematics works, experts say.

"It's a miracle," said Donald Grossman, a mathematician at Harvard University. "This was something mathematicians had proven beyond a shadow of a doubt to be impossible."

A Klein bottle, named after the late-nineteenth-century mathematician Felix Klein, is a surface made by gluing together the ends of a tube in a prescribed way. If the tube is pulled into a circular shape and the ends are attached, the result is a doughnut-shaped surface, which mathematicians call a torus. But if the ends are glued to each other in the opposite direction, with their orientations reversed, the result is a Klein bottle. The catch is that there is no way to get the ends aligned properly without cutting a hole in the tube and letting the tube cross itself-a fact mathematicians "proved" years ago.

But Gassner has now defied conventional wisdom by building a perfect Klein bottle that never passes through itself.

Gassner started with no goal of shaking the foundations of mathematics. Rather, he was idly playing with a spare length of rubber tubing from one of his dental machines while watching a re-run of "The Brady Bunch" on television, when he somehow built the elusive shape.

"I didn't know it was a Klein bottle, but I thought it looked pretty interesting," he said. Curious to know more about the shape, he showed it to neighbor Leonard Krimsky, a mathematician at Columbia University in New York.

"I immediately realized what it was, and I was simply flabbergasted," Krimsky said. "I believe firmly in the mathematical proof that building a Klein bottle is impossible. But, doggone it, there it is," he added, pointing to the Klein bottle.

Gassner said Parker Brothers, the famous manufacturer of games, had already approached him about marketing the Klein bottle as a toy, to be called the "Impossibottle." But because Gassner was not paying attention to what he was doing when he made the Klein bottle, he doesn't know how to make another. "It's unique," he said, stroking the Klein bottle reverently, if a trifle regretfully. "I feel privileged to have been the vehicle to bring this to the world."

The mathematical community has greeted the news with awe and amazement. On recognizing the Klein bottle late last week, Krimsky immediately dispatched an e-mail with the subject header "KB built!" to several colleagues. In the next few days, the message raced through mathematics departments around the world, and since then, mathematicians have been abuzz with the news.

Hundreds of mathematicians have flocked to Paterson, many on foot, to touch or just look at the Klein bottle. Some are suggesting that it has healing powers.

Carl Nelson, a mathematics professor at Saint John's University in Collegeville, Minnesota, walked more than 1,000 miles to see the Klein bottle. "When I heard the news, I knew I had to come," he said. "I haven't been able to prove a theorem in seven years. I thought, just maybe, if I could touch the Klein bottle, its splendor would somehow revitalize my mathematical powers."

Although some mathematicians are citing divine intervention, others have greeted this suggestion with skepticism, arguing that the Klein bottle is evidence instead that the world has more than three dimensions or is non-orientable, a counterintuitive property in which a right-handed object can be converted into a left-handed object simply by being passed along an appropriate path through the universe. In a non-orientable universe, bottles with no outside or inside are mathematically plausible.

"While Gassner was watching television, the natural motion of the Earth must have carried him through a small non-orientable pocket of the universe," said Boris Harkov, a mathematician at the Massachusetts Institute of Technology in Cambridge. "That's the only reasonable explanation."

One way to test the orientation of the universe is to hurl a right-handed glove into the air and see if it falls back to Earth as a left-handed glove--if it does, the universe must be non-orientable. Since Gassner's announcement, physicists have been carrying out such experiments, both outdoors and in Gassner's TV room, but so far all tests have come back negative. Still, many researchers are optimistic. "I'm confident that the glove will flip soon," said Chen Xiang, an experimental physicist at Brookhaven National Laboratory in New York.

Those who believe the Klein bottle is evidence of a higher deity and those who think instead that it means the universe is non-orientable agree about one thing: the discovery of the Klein bottle is a monumental step forward for mathematics.

"It's a simply astounding breakthrough," Grossman said. "This discovery ushers in a new era of mathematics and physics."