Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow, The Free Press, $26, 306 pages.

The Chapel Hill News


June 27, 2001

Einstein's debt to geometry


By Phillip Manning; CHN Columnist

Geometry is a high school subject that has few practical applications and is not important in modern science. At least that's what I thought before I read Leonard Mlodinow's captivating book "Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace" (The Free Press, $26). It turns out I was wrong on both counts. Geometry has important practical applications, and it played a starring role in one of the great scientific achievements of the 20th century.

What, for example, could be more practical than measuring the size of the Earth? It was first done in 212 B.C. by Eratosthenes of Cyrene. He noticed that at noon of the summer solstice a stick stuck perfectly upright in the ground in his hometown cast no shadow. That meant that the sun's rays were parallel to the stick. If the earth is a sphere, Eratosthenes reasoned, then a stick in the ground a few miles away should cast a shadow. Eratosthenes then measured the angle of a shadow cast at noon in Alexandria. Employing perhaps the world's first graduate assistant (and treating him in a manner that all graduate assistants will understand), Eratosthenes had him walk between the two towns and measure the distance, a stroll that turned out to be 500 miles. Using the theorems of Euclid, the man who devised the geometry we study in high school, Eratosthenes calculated the circumference of the Earth. He came up with about 25,000 miles, very close to the correct value.

To illustrate geometry's relevance to modern science, Mlodinow considers the problem of gravity. Before Isaac Newton (1643-1727) came along, nobody understood much about the physical world. The medieval theologian Thomas Aquinas said that the sun rotated around the earth because the hand of God was pushing it. Only after Copernicus proved that the earth revolved around the sun and Newton developed his laws of motion and gravity did the movement of the planets make sense. But even Newton, arguably the smartest man who ever lived, was stumped about how gravity operates. The law of gravity states that two bodies attract one another and that the attraction depends on the masses of the two objects and the distance between them. Time plays no role in this law. How can a force be transmitted instantaneously?

Because Newton's laws worked so well, few scientists worried about this conundrum until an ambitious patent clerk in Switzerland published a paper in 1905. The writer was Albert Einstein, and the paper developed a new theory about how the world works. It was called special relativity, and it said (among other things) that nothing could travel faster than the speed of light. Newton had been right to be uncomfortable with the instantaneous transmission of gravitational force, and now Einstein's theory proved it to be impossible. How did gravity work?

After pondering the problem for seven years, Einstein came up with the answer. Gravity is not a force, but a curvature in the fabric of space. And if space is warped, Einstein concluded, Euclidean geometry will no longer hold. To finish constructing his theory, Einstein needed a new geometry, one that operated in curved space. In this strange universe, Einstein realized that the sum of the angles in a triangle is no longer 180 degrees and the Pythagorean theorem, a staple of high school geometry, is no longer true. But he didn't have a clue as to what mathematics ruled in such a space.

With the help of a friend, he uncovered the work of Georg Riemann, a geometer who had worked out the mathematics of curved space half a century earlier. Riemann was born in 1826 in a small village in Germany to a poor family, He seemed, Mlodinow writes, "a bit too smart to be one of us." When he was 19, a teacher lent him a copy of Legendre's "Theory of Numbers," 859 pages of abstract mathematical theory. Riemann returned the book in six days, pronouncing it "a good read." Riemann's specialty was differential geometry, which is the mathematics of curved surfaces. It is an esoteric subject, but it has one easily visualized application - if you have a globe handy. Madrid lies due east of New York, but the shortest way to get from New York to Madrid is not to travel east, but to follow a line that curves northeast then southeast. This is because our planet is not flat but spherical, and the shortest distance between two points is a curve, not a straight line.

Einstein's genius was realizing that space itself was curved. After he discovered Riemann's work, he used the mathematics of differential geometry to work out the field equations for the general theory of relativity. Thus, an obscure branch of geometry allowed Einstein to produce the most important breakthrough in our understanding of the universe since Newton developed his laws.
####