An Angle on Beauty

A mathematics historian and an artist team up for a show-and-tell about geometry, in a book that lives up to its name.

Beautiful Geometry. Eli Maor and Eugen Jost. 187 pages. Princeton University Press.

This Is Not the Square Root of 2.
This Is Not the Square Root of 2.

You don’t have to love mathematics or its history to appreciate Beautiful Geometry, but it definitely helps. Eli Maor’s lively writing benefits in equal parts from the geometry of ancient Greece and the eye-popping images conjured by artist Eugen Jost.

Among the many Pythagorean discoveries Maor highlights is the irrational number—specifically, the square root of 2 (infinite nonrepeating digits starting with 1.4142 that, when multiplied by itself, equals 2). Previously, all numbers were thought to be rational—a positive integer or a ratio of two positive integers. This introduction of a number that defied rational behavior triggered a 2,000-year rift, Maor tells us, in math’s two main branches: geometry and arithmetic. The plate above is called This Is Not the Square Root of 2 (after Magritte’s famous pipe painting and claim of its non-pipeness), because, as Maor explains, “the long string of decimals in our illustration is not the square root of 2, just a close approximation of it.”

Pythagorean Metamorphosis.
Pythagorean Metamorphosis.

Other illustrations represent what might be thought of as classical geometric depictions, like the color-shifting old-cigarette-lighter-like objects in Pythagorean Metamorphosis, which is a take on Euclid’s Proposition 47 in his first book of Elements. It’s also known as the Pythagorean theorem, but Maor notes that Euclid himself didn’t label it as such because he didn’t believe in calling out individuals. It was the geometry that mattered, Maor writes, and here we see right triangles (white) “whose proportions change from one frame to the next, starting with the extreme case where one side has zero length and then going through several phases until the other side diminishes to zero.” According to Euclid’s lemma, a preliminary result that posits “the square built on one side of a right triangle has the same area as the rectangle formed by the hypotenuse and the projection of that side on the hypotenuse,” the painting attests that “the two blue regions in each phase have equal areas, as do the orange regions.”

Prime and Prime Again.
Prime and Prime Again.

Jost twists many marvelous and palatable arrangements from spheres and squares and other basic shapes, but this reviewer’s eye keeps returning to a subset of illustrations that carry a strange gravity. These are the images made of numbers, after Robert Indiana or Jasper Johns, spun into an evocative pattern, like Prime and Prime Again, in which the top sequence sheds digits until only a 7 remains. “For no apparent reason,” Maor writes, “each number in this sequence is a prime.”

 

 

Bill Cannon is the editor of Scientists' Bookshelf. His writing has appeared in Nature, Smithsonian, The New York Times Magazine and American Scientist magazine.