9 April, 2009
The H&FJ Institute for Unapplied Mathematics
We've received our share of intriguing questions over the years, but this one takes the cake. On Monday, a correspondent called from National Public Radio to discuss the implications of typesetting a number with twelve million digits.
The number in question is 243112609-1, which holds the title for World's Largest Known Prime Number. Mathematicians have known since at least the third century BC that for many values of n, the formula 2n-1 produces a prime number. When it does, the result Mn is called a Mersenne Prime, after the seventeenth century French mathematician who calculated the first 257 of them by hand — quite something when you realize that M257 has 78 digits. (And, so very cruelly, it's not prime.) The search for prime numbers, an esoteric pursuit that rivals typeface design for its cultishness, has continued ever since; these days it's assisted by the Great Internet Mersenne Prime Search, a project that organizes the downtime of almost 90,000 volunteers' computers into a collective effort to find the next great prime.
Monday's call came from Joe Palca, a science correspondent for NPR's Morning Edition, who was meditating on the best way to convey the magnitude of this number. Scientific notation is designed to reduce astronomically large numbers down to more manageable ones, which obscures the enormity of numbers like 243112609-1. What does it mean for a number to contain 12,978,189 digits? For comparison, the total number of atoms in the universe is often estimated at 1080, a mere bagatelle of just 81 digits.
Joe liked the idea of measuring how long this number would be if it were set in type, which immediately called into question the choice of font. The number's length would depend chiefly on the width of the font selected, and even listener-friendly choices like Times Roman and Helvetica would produce dramatically different outcomes. Small eccentricities in the design of a particular number, such as Times Roman's inexplicably scrawny figure one, would have huge consequences when multiplied out to this length. But even this isn't the hairy part. Where things get difficult, as always, is in the kerning.
Kerning pairs mitigate the space between awkwardly fitted characters, and numbers contain one of the deepest kerns in a font: the pair 74, whose slanted profiles need to be specially accommodated. H&FJ's Andy Clymer, always up for a programming challenge, wrote a script that examined all twelve million digits (download them here), and concluded that "74" appeared 129,818 times. If we were to set our target number in 12pt Gotham Book, this kern alone would account for a variance of more than 150 feet. So to account for this and all the other 100 possible inter-number kerning pairs, Andy wrote a script that quietly chewed away on the big number with a handful of fonts in mind. The results?
|12 point Gotham Book||20 miles
|12 point Whitney Book||18 miles
|12 point Gotham Narrow Book||17 miles
|12 point Mercury Text||17 miles
Of course, the math gets quite a bit easier in fonts with tabular figures, where all of the digits are the same width, and as a bonus there's no kerning to deal with. (All of the above fonts feature tabular figures as well, but what fun is that?) In the end, we thought that a calculation in 12pt Courier would be most evocative to listeners, and the easiest for NPR fact checkers to confirm as well. So for the record, 243112609-1, set in 12pt Courier, runs to 20 miles, 2,569 feet, and 1-3/4 inches (32.96 km).
In case you're wondering, prime numbers aren't just the stuff of academic longhairs: like typefaces, they have interesting properties that make them strangely useful. The classical example comes from mechanical engineering, where two meshed gears will wear most evenly if each has a coprime number of teeth, since this evenly distributes the possible ways in which they interact (thereby minimizing the effects of any irregularities.) Some have suggested that 13- and 17-year cicadas each follow prime numbered life cycles in order to ensure that their populations compete as little as possible, coexisting only once every 221 years. As for the larger primes, the challenges of quickly factoring large numbers makes large primes indispensable for securing online data, so they're one of the cornerstones of public key cryptography. Large primes are acting behind the scenes when you're online ordering fonts, for example. When you're online ordering fonts. —JH