From Science News 20th May 1995 (I remember reading this note in Nature at the time)
(By the way Bob Matthews is the guy who proved that in any universe with physical
constants compatible with intelligent (carbon based?) lifeforms, Buttered Toast like objects
falling from a worktop at a height suitable for such lifeforms would fall butter(-oid) side
Spying pi in the sky
In the April 20 NATURE, Robert A.J. Matthews of the applied mathematics and computer science department at the University of Aston in Birmingham, England, describes how to use the distribution of bright stars across the night sky to deduce a surprisingly accurate value of pi, the ratio of the circumference of a circle to its diameter.
“My aim was to extract something mathematically interesting from something we’re all familiar with,” Matthews says.
This result follows from an application of certain theorems in number theory. These hold that, given any pair of whole numbers chosen from a large, random collection of numbers, the probability that the two numbers have no common factor other than 1 is 6/¹2 (about 0.61). For example, in a set of numbers including 8, 9, and 27, 9 and 27 have the common factor 3, whereas 8 and 27, as well as 8 and 9, have no common factor apart from 1. It’s possible to calculate the value of pi by determining what proportion of pairs of whole numbers selected from a large, random sample has no common factors.
As the source of random numbers for his “celestial” estimate of pi, Matthews used the angular separation between the positions of pairs of the 100 brightest stars. He checked a million pairs of these numbers for factors and obtained a value of 3.12772 for pi, which is within 0.5 percent of the actual value of 3.14159…. “The ancient Greeks used to believe that numbers lie at the root of all things,” Matthews notes. “I guess this result tends to support that idea.”
And here is the original:
Pi in the sky
Nature 374 681-682 1995
Over the years, people have pointed out that it's possible to get a pretty accurate value of the famous constant Pi ( = 3.14159....) from studying the dimensions of the Great Pyramid. Its base measures 230.35 metres, while its slanting face angle is 51.867 degrees, leading to an apex height of 146.71 metres, and thus a ratio of base to height of 1.5701, which is within 99.96 per cent of the true value of Pi/2. Some have even gone so far as to claim that this proves that aliens helped the great Egyptian engineer/architect Imhotep erect the pyramids. More likely, the presence of Pi in the design is really just a figment of the surveying techniques used, which may well have featured wheeled devices whose circumference is of course simply Pi times the wheel's diameter. In this paper, I show that it is possible to extract a similarly accurate value of Pi from a rather more "cosmic" source: the scattering of the brightest stars in the night sky.
Analytical number theory shows that if any two numbers are plucked at random from a large collection, the probability that two random numbers have no common factors apart from 1 is 6/(Pi)^2. Thus, by taking a large collection of random numbers and checking each pair for common factors, it is possible to estimate Pi. For a "cosmic" source of random numbers, I took the scattering of the 100 brightest stars in the night sky, worked out the angular separation of each star from every other star, and used this to create a collection of one million pairs of random numbers. By using the famous Euclidean algorithm for finding greatest common divisors, I then counted up the number of these pairs that had no common factor, and set the proportion equal to 6/(Pi)^2. The result leads to a value for Pi of 3.12772 - within 99.56 per cent of the true value. All of which shows, I guess, that whacky New Age ideas about mystical numbers and the cosmos aren't as whacky as they sound.
At the time this was published I was a bit perturbed by Bob's reuse of
the same stellar positions to generate a larger sample than possible if
each position is used once. However I simulated the procedure and it
IIRC did give better estimates than what I expected. Perhaps there is
a paper in justifying this reuse of points
Needles to say I like this approach as it is an example of what I call
Mad Science, which is also exemplified by Watson and Cricks work
on the structure of DNA.
The approach of Mad Science, and Mad Scientists is what attracted
me to science as a child, and is now under threat from what might
be called scientific political correctness, which holds that dull and boring
systematic carefully planned science (as exemplified by Jim Watson's
portrayal of Rosalind Franklin in the Double Helix) is correct science and
Mad Science is an aberration to be killed off.