An Unsung Geometer Keeps to His Own Plane

Drive into the mountains past Santa Barbara, past Lake Cachuma, and presently you come to the hilltop home of Herbert Busemann, who at the age of 80, 15 years after retiring, is still going strong as a mathematician, painter, linguist and individualist.

From 1947 to 1970 Busemann was a professor, then a distinguished professor, at the University of Southern California, where he developed and honed his theories of non-Euclidean geometry. He has just been named as the winner of the Soviet Union’s Lobachevsky Prize, the first American mathematician ever so honored by the Russians, who have been giving the prize every four years for more than a century.

Busemann will go to Moscow later this year to pick up the award and the 2,000 rubles that go with it.

Few mathematicians ever make it into public consciousness, but Busemann has had a hard time even within his own field, in part, at least, because he never worked on trendy problems and never followed the crowd. Until he got to USC, his academic advancement was slow. Despite substantial contributions to mathematics, he was never elected to membership in the National Academy of Sciences. “If I have a merit, it is that I am not influenced by what other people do,” Busemann says.

“Tastes change a lot and interests change a lot in the space of five years,” said Bob Brooks, a young geometer at USC. “But there are people who aren’t so interested in keeping up with today’s fads. He’s very definitely in that category.”

Busemann characterizes his basic mathematical approach this way: “Any apparently difficult problem can be done with very simple methods. This is the property of many of my things. I see a simple geometric reason which others have overlooked.”

Despite a lifelong desire to paint, Busemann never took it up, fearing that it would divert him from the arduous work of mathematics. But retirement freed him, and he built a studio in his home that is now chockablock with dozens of large canvases painted in vibrantly colored geometrical designs. His mathematical vision carried over into art, and some of the paintings have been sold.

Busemann says the painting stems from the same urge to create that spurred his mathematical work, which still goes on. “Does creativity have to be defined?” he asked. “It is the urge to do something new, to invent something. I can’t tell where it comes from. I don’t know whether it is curiosity or what.”

Busemann, born in Berlin in 1906, studied in Gottingen and Munich in Germany and in Paris and Rome before coming to this country in 1936, having decided that living under Hitler would be impossible. Along the way, he has known and worked with many of the eminent mathematicians and scientists of this century, including David Hilbert, Niels Bohr and his brother, Harald Bohr, John von Neumann and Albert Einstein, whom he met at the Institute for Advanced Study in Princeton, N.J., where Busemann spent three years after arriving in the United States.

Einstein showed some interest in Busemann’s geometry, hoping that it would be useful in developing the unified field theory that the great physicist unsuccessfully sought in the last decades of his life. After a while, though, it became clear that that approach would not work.

One characteristic of Einstein stood out to Busemann. “If the total population of the world consists of n people, and n -1 people are of one opinion and he is of another, you cannot shake him,” Busemann recalled. “He was completely independent of what anyone else might think.”

Having lived and worked in many places, Busemann acquired many languages and taught himself others as needed. He has lectured in seven languages--German, English, French, Spanish, Italian, Danish and Russian--and his home is filled with books in many of them. He also keeps up in Latin and Greek, which he studied as a child. “Every two years I read the ‘Odyssey,’ I like it so much,” he said. “And Plato.”

Now about the mathematics. More than 2,000 years ago, Euclid invented geometry, the fifth postulate of which states that through a point not on a line, one and only one line can be drawn parallel to the given line. These parallel lines will never meet, even if extended to infinity.

From the beginning, people were troubled by this postulate because, unlike the others, it is not verifiable. There is no piece of paper of infinite length on which the lines can be extended. For two millennia, geometers worried about Euclid’s Parallel Postulate, trying unsuccessfully to prove it in a variety of clever ways.

Finally, in the 19th Century, the Russian Nikolai Ivanovich Lobachevsky and the German Bernhard Riemann replaced the Parallel Postulate with other postulates and invented non-Euclidean geometry. In Lobachevsky’s scheme, infinitely many lines may be drawn parallel to the given line, and in Riemann’s, no lines may be drawn parallel to the given line. It turns out that both of these geometries are logically equivalent to Euclid’s. They are all logically consistent, and none is better or worse than another.

As with many discoveries in mathematics, non-Euclidean geometry began as a purely intellectual exercise. Later it wound up having practical applications. The curved space-time of Einstein’s universe happens to be non-Euclidean. Why mathematics, which is a creation of the human mind and imagination, correlates with the physical world is a deep philosophical mystery.

In any case, Busemann’s contribution, developed in detail in his 1959 book, “The Geometry of Geodesics” (Academic Press), was to develop a method for dealing with a non-Euclidean plane. He emphasizes that the approach is more important than any particular results, though there have been results. “The emphasis is more on the radically new approach than on the individual problem,” he said.

So far, there has been only one practical application for this work, and that was short-lived. During World War II, submarine captains found that their torpedoes systematically missed their targets. Why? Well, on the open sea, looking through a periscope without reference marks for comparison, the view is non-Euclidean.

This discovery was made a half year before radar was invented. “Radar was better,” Busemann said.