On this clear blue, purified spring day, Dan takes a postprandial stroll and Manhattan becomes a three-dimensional chalkboard. Between the geometry of architecture and calculus of urban life, you begin to see the sidewalks and the skyscrapers through a mathematician's eyes and somewhere along the way, theoretical math becomes, well, more concrete.
A sunflower at a florist's shop helps illustrate Fibonacci numbers. A stack of tomatoes at a greengrocer suggests Kepler's Conjecture. A stand of seven trees leads back around to a conversation about the Riemann Hypothesis. It's like taking a tour of a familiar place with a foreign-tongued guide.
Things You Can Count On
Dan Rockmore has long been enamored of numbers. He remembers as a little boy, walking with his physicist father to buy a newspaper in the morning. He talked to his dad about fractions; sometimes he counted his steps. Decades later he speaks and writes of math -- and the beauty of proofs -- with the reverence of a loving son. "We count. That's what we do," he says. "I have always assumed that everybody did it."
We are also, he says, hemmed in by math. "Life is about being creative within bounds," Rockmore says. "You can be infinitely creative, but there are some hard and fast rules." Often those rules are represented by numbers. You only have two hands. There are only 24 hours in a day. The alphabet has 26 letters; the major musical scale seven notes.
In "Stalking," he writes that the natural numbers -- the plain old numbers we use every day, such as one, two, three, four, etc. -- seem to have been with us from the beginning of time. They "are implicit in the journey of life, which is a nesting of cycles imposed upon cycles, wheels within wheels. One is the instant. Two is the breathing in and out of our lungs, or the beat of our hearts. The moon waxes and wanes; the tides ebb and flow. Day follows night, which in turn is followed once again by day. The cycle of sunrise, noon and sunset gives us three. Four describes the circle of the seasons."
Though you won't have to count much in this article, you must understand that there are two types of natural numbers: composites and primes. Composites, such as 4, 6, 8, 9, 10, 12 and so on, can be divided by smaller numbers. The primes -- 2, 3, 5, 7, 11, 13, 17 and so on -- cannot be divided by smaller numbers, except 1. Prime numbers have a practical application these days; we use them in e-commerce to encrypt digital information, which makes it harder for identity thieves to steal our Social Security and credit card numbers on the Internet.
Mathematicians were intrigued by the primes long before there were computers. Euclid, a 3rd-century B.C. mathematician in Greece, pointed out that there are an infinite number of primes. Leonhard Euler, an 18th-century Swiss mathematician, discovered that primes appear in certain series. C. F. Gauss, an early 19th-century German known as "the prince of mathematics," tried to figure out why the primes are farther and farther apart as you count higher and higher. And in 1859, German genius Georg Friedrich Bernhard Riemann put forward a hypothesis that prime numbers occur -- along that never-ending number line -- in a certain pattern. He concocted a formula that helps to predict when the next prime will occur. Today it is called the Riemann Hypothesis.
It has not been proved, beyond the shadow of a doubt, that Riemann's formula can predict the primes all the way out to infinity. That is why mathematicians still fiddle with the hypothesis and why Rockmore wrote his book.
Pointing to a sparkling yellow-and-black sunflower at Apple Tree Flowers on the corner of 69th Street and Second Avenue, Rockmore speaks of Leonardo Pisano Fibonacci, a 13th-century Italian mathematician. Fibonacci is famous for figuring out a special sequence of natural numbers. He began with 0, then 1. From then on, he added the previous two numbers to find the next. For example, 1, 2, 3, 5, 8, 13, 21, 34. A simple-enough pattern -- called the Fibonacci series -- but it becomes profound when you discover that the numbers pop up throughout the natural world: in certain flower petals, pine cones and the seed head of a sunflower, where the number of spirals -- usually 34 or 55 or 89 -- allows nature to pack as many seeds as possible into a circle.
A few blocks away, Rockmore pauses in front of a display of vegetables at the Garden Deli. The tomatoes are stacked neatly, one layer latticed atop another. Mathematicians marvel at the ways spheres fill up space, he says. Johannes Kepler asserted in 1611 that this way of stacking -- called "face-centered cubic packing" -- is the most efficient, but for centuries no one could prove it. A proof is a very detailed, logical process for verifying a mathematical assertion.
In 1998 Thomas Hales at the University of Pittsburgh posted a proof of the Kepler theorem online that, assuming that one is dealing with perfect spheres and perfect cubes, was eventually accepted by the math community.
The Green Space Theorem
To really contemplate natural numbers and everyday math, Central Park is the place to go. Historically, great math problems and solutions have sprung from walks in a park. Euler, for example, pondered a famous conundrum while strolling through the parks of Konigsberg. Two rivers flowed through the East Prussian town, among two islands and the mainland. All in all, there were seven bridges connecting three pieces of land. Townsfolk made a game out of trying to cross all seven bridges during a single walk -- without backtracking.
