Happy Pi Approximation Day!
Pi, as everyone but samuraipandapoetry will recall, is the ratio between a circle’s circumference and its diameter. If the diameter of a circle is 1, than its circumference is π. Its value is traditionally represented as 3.14159… Some people celebrate the 22nd of July for its approximation of π (22/7). Others celebrate March 14 (3.14), although the deluded followers of the March date refer to it simply as Pi Day. But all values for π are approximations, because π is irrational. The ancients did not know this - and it wasn’t finally proven until the 18th century - but π cannot be expressed as any exact fraction. (This despite the Indiana House of Representatives passing a law stating that π = 3.2.) Pi extends forever and ever, and where π goes, its devotees follow.
In physics, π appears in the cosmological constant, Heisenberg’s uncertainty principle, and Einstein’s field equation for general relativity. Pi comes at us in unexpected ways, too, often having nothing to do with circles. It arises in the distribution of prime numbers, and in the probability that a pin dropped on a set of parallel lines will intersect one of them. The ratio of the actual length of a river and its “as the crow flies” length is, on average, π.
An Egyptian named Ahmed provided a fairly accurate estimation for π around 1650 B.C. He approached the problem as one of “squaring the circle,” that is, constructing a square with the exact same area as a circle. The “constructability” problem would bedevil mathematicians for three and a half millennia until it was proved unsolvable. Antiphon and Bryson of Heraclea in the 5th century B.C. worked on calculating the area of a circle and invented the “principle of exhaustion.” Calculating the area of a polygon is trivial, so Antiphon repeatedly doubled the sides of a hexagon inscribed in a circle, so that it came closer and closer to approximating the circle. Bryson’s insight was to add another polygon, this one inscribed around the outside of the circle, and to treat the solution as one of lower and upper bounds.
Two hundred years later, Archimedes (of “Eureka!” fame) devised the first known theoretical method for calculating π. Building on Antiphon and Bryson’s exhaustion method, he focused on the perimeters of polygons instead of their areas, thereby estimating the circumferences of circles. Using 2 96-sided polygons, he calculated π as 3.1419, which is within three ten-thousandths of its actual value.
In 263 A.D. Liu Hui in China used polygons of 3,072 sides to come up with an estimate of 3.1416. Later scientists would use polygons of 24,576 sides and even 393,216,216 sides to calculate π accurately to 10 decimal places. In the 16thcentury, the Dutchman Adriaen Romanus used a polygon with over 100 million sides to achieve 15 digits, and the German Ludolf van Ceulen used polygons with 32 billion sides each to calculate π to 20 places. van Ceulen later reached 35 decimal digits, and today in Germany π is sometimes called the Ludolfian number.
By this time, the exhaustion method had been pretty much exhausted. Brute force gave way to more elegant methods such as infinite product series and the Gregory-Leibniz arctangent series. These are “converging” methods, where subsequent terms of a formula hone in on a true value. One drawback of these methods is that they can be maddeningly slow. The original Gregory-Leibniz series required 300 terms to extract a meager 2 decimal places of π, and would have required more terms than there are particles in the known universe to achieve 100 decimal places (by which time you would probably run out of paper anyway). Nonetheless, once refined, this method forced π to yield over 500 of its precious digits by the nineteenth century.
The next major breakthrough for calculating π came with the invention of electronic computers and yet more efficient algorithms. Suddenly, thousands of digits were being calculated, and records were being overturned more and more rapidly. In 1973, 1 million decimal places of π were computed. By the 1980s, records of 16 million and then over 201 million digits had been set. In 1989, David and Gregory Chudnovsky found 480 million digits using a home supercomputer they custom built from off-the-shelf parts that took up most of their apartment. Later that year, they reached 1 billion digits. When a competitor reached 6 billion digits 6 years later, the Chudnovskys turned around and attained 8 billion digits the year after that. In 1997, a Japanese team calculated 51.5 billion digits of π. The same team set the current record of just over 1 trillion digits.
Make no mistake: 30 decimal places would suffice to measure a circle the size of the observable universe to within the width of an atom. Calculating decimal places beyond that has no practical value. The strenuous, sometimes life-long exertions of number theorists throughout history to eke out the next digit of π, and then the one after that, have served the goal of finding patterns in the unending river of numbers. The mystery is the message. But as far as anyone has been able to determine, the digits are more or less random.
If mathematicians are the knights errant of π, those who attempt to memorize π are its troubadours. Simon Plouffle memorized over 4,300 digits in the 1970s. Rajan Mahadevan set a world record in 1983 by memorizing 31,811 digits. In 1995, Hiroyuki Goto memorized 42,000 digits. The Guinness Book of World Records recognizes Lu Chao for reciting 67,890 digits of π from memory.
I’ll leave you with this thought. Some people believe that folks born on this day are unusually good-looking, talented, and witty, and that they are exceptionally skillful lovers as well. [Um, I’m going to need to review your notes on this. – Ed.]
Source: David Blatner, The Joy of π, Walker & Company, NY, 1997 (except where otherwise indicated).
Image of pi/pie via http://en.wikipedia.org/wiki/Pi
Image of Archimedes via http://www.math.nyu.edu
Image of π poster via http://www.unihedron.com/projects/pi