Raphael Rosen - Math Geek: From Klein Bottles to Chaos Theory, a Guide to the Nerdiest Math Facts, Theorems, and Equations

Avon, Massachusetts

To Nathaniel, Jolina, and the rest of my family

67. How Many Clues Do You Need to Solve a
Sudoku Puzzle?

WHAT DOES IT MEAN TO BE A MATH GEEK? Maybe you enjoyed math classes when you were in school and now do logic puzzles in your spare time. Maybe youve become intrigued by all of the references to math in popular cultureProof, Numb3rs, The Imitation Game, A Beautiful Mindand want to know more. Maybe youre an engineer or a physicist and use advanced math concepts every day. Maybe you have a hard time understanding math and yearn to get a glimpse of a world that so many people find fascinating. Or maybe youre your own kind of geek: after all, there are as many kinds of math geeks as there are theorems.

Whoever you are, in these pages I hope to show you that mathematics is not just a series of rote exercises performed in a classroom. You wont have to memorize anything, and theres no test at the end. I hope to convince you that mathematics is something built into the fabric of reality: a collection of shapes, patterns, numbers, arguments, and, well, little treasures. Math is in the air you breathe, the sidewalks you walk on, and the buses you take to work each morning. What does that mean? To find out, youll have to read on.

Besides showing you that mathematics is a living feature of the world we live in, I also hope to persuade you that math is pretty. Im not saying that equations look good on paper, or that plus and minus signs are like calligraphy. I mean that learning about math is like looking at a sunset, reading a poem, or listening to your favorite band. Math has a beauty that can stop you in your tracks. Have you ever walked out of a movie theater after seeing a great drama, your mind stunned by the performances, sets, and cinematography? Believe it or not, math is like that. Some mathematicians have even argued that math should be included in a list of cultural touchstones that includes Shakespeare, Mozart, and Michelangelo. These math mavens believe that people should study math for its own sake, because not studying it would be a crime on a par with never having read Hamlet. In other words, people should not learn about mathematics simply to get a good score on the SAT. Instead, they should study it to enrich their lives.

Our expedition to find math in our everyday world will take us from pizza to donuts, from online shopping to the GPS features in our smartphones. Well look closely at why, when youre waiting at a bus stop, there wont be a bus for what seems like an eternity, and then suddenly two or three will arrive all at once. Well stop to examine weird vegetables in your local supermarket, and learn how music is translated into a file on your iPod. Well even make sense of strange paradoxes like why adding roads can make traffic worse.

Once you learn the fascinating math concepts hiding in the world around you, youll find an even greater appreciation for mathone that you can share with your fellow commuter when the bus is late... again.

HAVE YOU EVER looked closely at the fruits and vegetables in your local supermarket? Some of them are creepy: the yellow Buddhas hand, for instance, looks like a squid creature from an H.P. Lovecraft story. Others are strangely beautiful. Sweet potatoes have a wonderful lumpiness like misshapen hunks of clay; onions have nesting rings like those found in trees; and when you slice open an apple, you can see that the seeds are arranged in a star-shaped configuration thats oddly pleasing. Even ornamental cabbagewhich is sold at garden storeshas a kind of geometric attractiveness.

But nothing in the produce aisle beats the Romanesco broccoli for the ultimate in vegetable beauty. In fact, its hard to take your eyes off it. The Romanesco, a type of Brassica oleracea, or cabbage, has the general shape of a pinecone, but its surface is a riot of smaller pinecone shapes, and on the surface of each of those cones are more cones, and so on. Each of the smaller surface cones looks like the larger, original cone, so much so that if you were to lop off a surface cone, photograph it, and place the photo next to an image of the entire broccoli, it would be difficult to tell which was which.

Mathematicians would say that the shape of the Romanesco is self-similar: if you zoom in on the shape and look closely at a detail, what you see is the same as if you hadnt zoomed in at all. Self-similaritywhen an object looks the same no matter the scaleis also a distinguishing feature of fractals, a kind of shape studied and popularized by mathematician Benoit Mandelbrot. His 1982 book, The Fractal Geometry of Nature, helped introduce this species of object to the world. (The book was largely a revision of his 1977 book called Fractals: Form, Chance and Dimension.) Mandelbrot identified lots of shapes in nature that have this self-similar quality: jagged coastlines, clouds, and the exquisite tracery of veins in a leaf. Nature seems to like self-similar shapes; the more you look for them, the more you find.