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Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math

Free Press
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Alex Bellos’ new book is one that, at first glance, seems amazingly trivial. The mathematics is anything but deep. But put it down, mull over the ideas, and the book takes on a new dimension that is both delightful and fun. It’s a book that is best read when you are in a mood to tour mathematics for a cursory look of its many applications and uses. There is nothing complicated or esoteric, just a simple tour of numbers, geometry, and even modern day research.

Let’s look at some highlights.

We begin with counting and meet the Munduruku, a people who live in the Brazilian Amazon. The Munduruku can count, but only up to three: one, two, three, many. (In The 4% Universe, Richard Panek extends this idea to our universe: “Earth, planets, Sun, far.”) They have no word for four, ten, or twenty. The live their lives without need of numbers or even a method to mark time.

Our next stop is geometry and who better than Pythagoras to see along the way. Bellos tells us the Pythagorean theorem and gives us a few proofs. We are then off to meet Euclid and learn that there are only five Platonic solids among the many other ideas of Euclid.

A short detour takes us to the world of origami. Origami, an ancient art, has only recently been studied for its mathematical beauty. For example, there is Haga’s Theorem: Fold a square sheet paper so that the lower right corner meets the top middle of the top edge. You will create three right triangles, actually Egyptian triangles, with sides in the proportion of 3:4:5.

Our tour now goes to India. Bharati Krishn Tirthaji is a Hindu holy man. At the age of 82 he visited the United States and lectured at the California Institute of Technology on arithmetic. He demonstrated, for example, how to multiply 9 times 8 without a multiplication table. That doesn’t sound so exciting but the text shows the steps and they are quite different than what is taught in elementary school.

Any tour of mathematics would be incomplete without π, and Bellos devotes a chapter to this marvelous number. We see π in circles, as the limit to sums, and we find that π has been computed to over 2 trillion digits.

From π we go to magic squares, the arrangement of consecutive integers in a square matrix so that each row, column, and diagonal have the same sum. These lead us to the Sudoku puzzles of today and then to Tangram puzzles of old.

Our tour then takes us to meet Neil Sloane, who collects sequences of integers. He runs the Online Encyclopedia of Integer Sequences ( and has cataloged over 160,000 entries. We see a few of these sequences and we learn how to (possibly) create our own.

After being around various parts of the world from Brazil to India and places in between, our tour ends with a brief look at the spaces of Euclidean and Non-Euclidean geometry.

Bellos takes us through many old, new, and useful areas of mathematics. His style is playful, light, and non-technical. He has a gift for explaining topics in easy to understand ways and the book is filled with diagrams and pictures that guide the reader along.

But remember: if you don’t like what you see the first time, take a break and come back to this book, you’ll be glad you did.

David S. Mazel received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.

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Friday, July 2, 2010
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Thursday, March 10, 2011