Calculating π

In the second season Star Trek episode “Wolf in the Fold” (1967), Captain Kirk and Mr. Spock force an evil entity out of the starship Enterprise’s computer by commanding the computer to “compute to the last digit the value of π”, thus sending the computer into an infinite loop. There is of course no last digit of π. In October 2014, a multi-threaded program called y-cruncher was used to derived π to 13.3 trillion digits (achieved by anonymous user “houkouonchi”) – it took 208 days on a system with two Xeon E5-4650L processors (2.6 GHz). Compare this to Ludolph van Ceulen (1540–1610), a German mathematician who spent most of his life calculating π. He managed 35 digits:

3.14159265358979323846264338327950288

The history of π encompasses many centuries. In the ancient world the value of π was frequently taken to be 3. A Babylonian clay tablet also gives a value of 3 -1/8 or 3.125 while the Rhind Papyrus which is dated 1650 B.C. gives a value of 4(8/9)2 or 3.16. The first attempt to compute π seems to be due to Archimedes of Syracuse (c.287 – 212 B.C.) who used a sequence of regular polygons inscribed in and circumscribed about a circle. The perimeters may be used to give lower and upper bounds for the value of π, and as the number of sides increases these bounds give better and better estimates. Beginning with regular hexagons and doubling the number of sides successively, Archimedes computed the perimeters of inscribed and circumscribed polygons with 6, 12, 24, 48 and 96 sides. For a 96-sided polygon he found that

3-10/71 < π < 3-10/70

which is equivalent to a value for π of 3.14 to two decimal places.

In 1942 ENIAC (Electronic Numerical Integrator and Calculator) was born, a “giant brain” in its time, capable of 5000 additions a second. It is hard to compare the “speed” of these antiquated systems with modern systems, but a good way is to compare the time taken to perform various calculations. ENIAC calculated 2037 digits in 70 hours in 1949. The Apple II, with 16kb of RAM performed the same calculation in 40 hours in 1978. No doubt the same algorithm now would run in a couple of microseconds. Computing has come along way in a little over six decades.

The question of course is why do we need to calculate pi to such accuracy? Partially as a benchmark. It takes a lot of effort to crunch numbers, and is a good indicator of how well a system can deal with a certain algorithm. Accuracy? Unlikely. The International Space Station Guidance Navigation and Control (GNC) subsystem performs calculations using 15 digits of π. Calculating π to 39 digits allows you to measure the circumference of the observable universe to within the width of a single hydrogen atom.

So why do humans strive to calculate as many digits as possible for π? Not because of any practical application – but rather to push the boundaries or what can be calculated, which is no different than trying to break the land-speed record.

 

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