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05/09/95

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Mathematician describes shape of the visible universe

STANFORD -- "I'm going to start with the punch line," Robert Osserman told his audience. Putting up a viewgraph that showed a shape something like a turnip, he declared, "That's the shape of the visible universe. Of course, now that you've seen it you might have some questions, like, 'How do you know? Or is this just a guess?' "

Thus, the professor emeritus of mathematics began the Karel de Leeuw Memorial Lecture on May 5. Its title was simply "The Shape of the Universe." That probably should have been the title of his popular book The Poetry of the Universe: A Mathematical Exploration of the Cosmos, recently released by Anchor Books.

Despite its title, the book doesn't contain much poetry, in the traditional sense.

"I didn't have a good working title for the book. Then my editor started calling it 'Poetry of the Universe.' We had a long debate over the title, but finally I went along with him," Osserman said.

The small volume does what one reviewer has called an "elegant" job of introducing its readers to many of the beauties of mathematics, specifically as they have related to people's efforts to understand the shape of the world and the cosmos. The basis for the book was Osserman's experience teaching one of the most successful interdisciplinary courses on campus -- "The Nature of Technology, Science and Mathematics" in the Values, Technology, Science and Society program -- that he developed and team-taught with applied physics Professor Sandy Fetter and mechanical engineering Professor James Adams.

In both the book and the lecture, Osserman began by describing early efforts to determine the shape of the Earth. Here the key concepts were those of map and curvature, he said.

One of the important types of world maps dates back 1,000 years and was invented by al-Biruni, who came from what is now known as Uzbekistan. This map is the type that is pictured on the seal of the United Nations; cartographers call it an azimuthal equidistant projection. It is very accurate near the center, but becomes increasing distorted at the outer edges.

Al-Biruni's map, and more familiar charts like Mercator projections, illustrate something proved by the great 18th-century mathematician Leonhard Euler: There is no such thing as an accurate map of the Earth's surface. That is the case, Osserman says, because the Earth's surface is curved and a map must be flat. If the world were a cylinder or even a cone, it would be possible to make a perfect map.

Three hundred years after al-Biruni, the Italian poet Dante created a verbal map of the universe, Osserman said. He put the Earth at the center of the visible universe, which consisted of a series of concentric spheres. Next he proposed a parallel, spherical universe, called the empyrean, where angels and other spiritual beings existed.

"The hard thing to imagine is that Dante envisioned the empyrean as simultaneously adjacent to and enveloping the visible universe," Osserman said.

Mathematically, it is possible to combine the inside of the two spheres to form a single whole. That is because these are maps, not pieces of the actual universe, and maps carry inevitable distortion.

The source of this distortion is curvature. It was the great 19th-century mathematician, Bernhard Riemann, who suggested that space itself might be curved, and proposed methods to measure that curvature. Much of Albert Einstein's work in this area is based on Riemann's ideas, Osserman said.

Carl Friedrich Gauss, another 19th-century great, realized that it was possible to determine the shape of a curved surface by making exact measurements of its surface. One way to do this is to start at a point and measure the circumference of circles centered on the point at successive distances from it. If these distances are less than that of a circle with a radius of the same length, then the surface has positive curvature, like the earth. If the distance on the surface is larger, then the curvature is negative, like that in the center of a saddle.

Riemann took these ideas and applied them to space itself. If space has a constant positive curvature, then a second portion of the universe, comparable to a first portion centered at the Earth, might be a kind of mirror image of the first, just as one hemisphere is a mirror image of the other.

In 1922, the astronomer Vesto Slipher presented the first evidence that the universe was expanding. Then, in 1929, Edwin Hubble presented further evidence for what is now known as "Hubble's Law," which states not only that the universe is expanding but that the further objects are away from the Earth, the faster they are receding. Taking this picture, and running it backward in time, has led to the "Big Bang" hypothesis for the creation of the universe, that it began not more than 20 billion years ago.

This would seem to be contradicted by the fact that, if one draws a series of concentric spheres moving out from earth, the earliest such sphere would seem to have the largest circumference. But this is also the oldest part of the universe, when everything was much closer together than at present.

This apparent contradiction goes away when you realize that these are maps of a curved universe and so contain distortion, Osserman said. While the direction and distance of these objects may be correct, the distances between them are probably exaggerated. In fact, the distances between these ancient objects must be much smaller than the distances that astronomers map with their telescopes.

Returning to Osserman's "punch line," an axis drawn through the center of the turnip represents time. The turnip's tip is the present day. As one moves down the axis, or back in time, the visible universe grows larger and larger, just as a turnip does, until you reach a point where the curvature of space causes such a large distortion that the visible universe starts to shrink back to the single point at which it began.

For a clearer, and much more complete explanation, read The Poetry of the Universe. Or listen to Osserman talk at the Stanford Bookstore this coming Thursday at 4 p.m.

-dfs-

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