Domokos and V?rkonyi used mathematics to design this self-righting object.

A software engineer in Virginia named Mike Keith wrote a poem to pi, a "piem." A love letter, in a way. To say that it is a something to behold is an understatement: It is nearly 4,000 words long ? and the length in letters of each word corresponds to pi's digits.

In other words, if you can remember the poem ? which riffs on T.S. Eliot's "The Love Song of J. Alfred Prufrock," Act V of "Hamlet" and Edgar Allan Poe's "The Raven," among other texts ? you too can recite pi.

"One: A Poem: A Raven," it begins (3-1-4-1-5). "Midnights so dreary, tired and weary, silently pondering volumes extolling all by-now obsolete lore. During my rather long nap ? the weirdest tap! An ominous vibrating sound disturbing my chamber's antedoor."

And so on.

Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .

This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).

Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.

Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.

Chapter1

This text introduces a new and simplified approach to trigonometry and a major restructuring of Euclidean geometry. It replaces cos, sin, tan and all those other transcendental trig functions with rational functions and elementary arithmetic. It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'. And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculus.

Notable Integers

-40 -1 222 255 273 360 369 496 666 720 880 1,001 1,729 2,520 142,857 275,305,224 2,147,483,647 4,294,967,297

Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., has a bicycle with square wheels. It's a weird contraption, but he can ride it perfectly smoothly. His secret is the shape of the road over which the wheels roll.

Stan Wagon rides his square-wheeled trike over a special roadway.
Courtesy of Stan Wagon

A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability , much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1996).

Benford's law applies to data that are not dimensionless, so the numerical values of the data depend on the units. If there exists a universal probability distribution P(x) over such numbers, then it must be invariant under a change of scale, so (1)

If , then , and normalization implies . Differentiating with respect to k and setting k = 1 gives (2)

having solution . Although this is not a proper probability distribution (since it diverges), both the laws of physics and human convention impose cutoffs. For example, if street addresses are distributed uniformly over the range of 1 to some maximum cutoff value, then they'll obey something close to Benford's law.

In 1986, puzzle enthusiast Lee Sallows of the University of Nijmegen in the Netherlands introduced a new form of magic square. He found magic squares for which the number of letters in the word for each number in a magic square generates another magic square.

Here's an example. Start with the following magic square, in which each of the rows, columns, and diagonals adds up to 45:

5 22 18 28 15 2 12 8 25

Spell out the English word for each of these numbers to obtain the following array:

five twenty-two eighteen twenty-eight fifteen two twelve eight twenty-five

Count the number of letters in each word and enter that number in the appropriate space of a blank three-by-three grid.

4 9 8 11 7 3 6 5 10

The result is another magic square

Coalitions drive down the number of ideal judges. "Suppose we have two judges who always vote the same way," Sirovich says. "Then, from the point of view of information, we have eight justices, not nine."

During the past 8 years, Justices Antonin Scalia and Clarence Thomas voted the same way more than 93 percent of the time, and Justices Ruth Bader Ginsburg and David Souter voted the same way more than 90 percent of the time. The fact that the number of ideal judges is as high as 4.68 is encouraging, Sirovich says.

Sirovich's work is an interesting analysis, Poole says. However, he cautions, many other studies suggest that the justices are heavily swayed by political viewpoints. "Only about 9 percent of their choices aren't explained by a simple liberal-to-conservative ordering," he says. "The court is very ideological."

Periodical cicadas usually have 13- or 17-year life cycles. Their development is so synchronized that practically no adults are present in the 12 or 16 years between emergences. When these cicadas do come out of their underground homes, they appear in huge numbers and create a cacophonous, throbbing din during their brief period of mating frenzy in the open air.

Curiously, 13 and 17 are both prime numbers, evenly divisible only by themselves and 1. The fact that periodical cicadas emerge after a prime number of years could be just a coincidence. Or it might reflect some sort of evolutionary pressure that leads to prime-number cycles.

In the early 1990s, working in the United States, Perelman had emerged as a major player in Riemannian geometry, which studies subjects such as curvature. "In that domain he was considered a phenomenon at that time, incredibly brilliant," recalls Cheeger.

Then abruptly, Perelman all but vanished from the mathematical scene. In 1995, he turned down job offers from several top universities and returned to Russia. When U.S. mathematicians asked Perelman's colleagues at the Steklov Institute what he was working on, they generally replied that they had no clue.

Some mathematicians speculated that Perelman had quit mathematics. Every now and then, however, one or another mathematician would receive an e-mail from Perelman with probing, insightful questions. "All of a sudden, there would be concrete evidence that he was following certain developments," Cheeger says.

Once Perelman's first paper on the Ricci flow appeared on the Internet in November 2002, rumors started flying that he had proven the Poincar? conjecture and Thurston's geometrization conjecture. On March 10, Perelman posted a second paper that developed the ideas in his first paper and explicitly claimed a proof of the two conjectures. He has promised a third paper with a few remaining details.

Can you guess the rules for generating these sequences?

In finding coin denominations that minimize the average cost of making change, Shallit assumed that every amount of change between 0 and 99 cents is equally likely. For the current four-denomination system, he found that, on average, a change-maker must return 4.70 coins with every transaction.

He discovered two sets of four denominations that minimize the transaction cost. The combination of 1 cent, 5 cents, 18 cents, and 25 cents requires only 3.89 coins in change per transaction, as does the combination of 1 cent, 5 cents, 18 cents, and 29 cents.

U.S. quarter.

"We would therefore gain about 17 percent efficiency in change-making by switching to either of these four-coin systems," Shallit says. "The first system possesses the notable advantage that we only need make one small alteration in the current system. We could speed up customer service just by replacing the dime with an 18-cent piece."

Whirled White Web: An award-winning, ill-fated snow sculpture.

Graham's number cannot be expressed using the conventional notation of powers, and powers of powers. If all the material in the universe were turned into pen and ink it would not be enough to write the number down. Consequently, this special notation, devised by Donald Knuth, is necessary.

SPIRAL SPREAD. One way to visualize the distribution of prime numbers is to arrange the integers in a square spiral, starting with 1 at the center of a grid, and then color the squares containing primes. The first grid (above) shows primes (red squares) among integers from 1 to 121. The second grid (below) shows primes (white or red squares) among integers from 1 to about 65,000. For more: http://www.sciencenews.org/20020504/mathtrek.asp.

You can easily generalize Fibonacci numbers to those that arise when each consecutive number is the sum of the four numbers that precede it. Ergo, tetranacci numbers: 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, and so on. In this case, the ratios of consecutive numbers tend to the constant 1.92756. . . .

Proceeding in a similar fashion, you can also generate sequences of pentanacci numbers, hexanacci numbers, and so on. Interestingly, as the number of added terms for generating a Fibonacci-like sequence increases, the constant that arises from taking the ratios of consecutive terms gets closer and closer to 2.

Legendre gave an estimate for (n) the number of primes n of (n) = n/(log(n) - 1.08366) while Gauss's estimate is in terms of the logarithmic integral (n) = (1/log(t) dt where the range of integration is 2 to n. You can see the Legendre estimate and the Gauss estimate and can compare them.