Spying Pi in the Sky

13. Pi in the Sky

Back to the Alien Digits

After the pie feast (see "Pi in the Sky"), the Digits invite Anita, Bill, and you to spend the night with them camping under the stars.

NASA

Soon the sky grows dark, and the Digits give each of you a sleeping bag. They line up their own sleeping bags in the same order they had sat at the picnic table: 3 1 4 1 5 9 2 6 5…. You and your friends set your sleeping bags nearby. Then, lying comfortably on your backs, all of you gaze up at a sky full of stars.

"Time to look for pi in the sky," says Three, handing each of you an object that looks like the protractor you use in geometry class for measuring angles.

"Now, pick any three bright stars," says One. "Use your protractor to measure the angle they form, like this."

"Now tell us the measurement of your angle, to the nearest degree," says Four.

Positioning your protractor, you measure an angle formed by three stars.

"Twenty-two degrees," you report.

Anita reports,"Ninety-eight degrees."

"Mine is 123 degrees," says Bill.

"Good," says One. "The numbers 22 and 98 are both divisible by 2. The numbers 22 and 123 have no common factor except 1. The third pair of numbers, 98 and 121, also has no common factors.

"So what?" asks Bill. "What do star angles have to do with pi?"

Spying Pi in the Sky

The Digits plan to estimate pi using a method introduced by British scientist Robert A.J. Matthews in 1995.

The method is based on the fact that for any two whole numbers chosen from a large, random collection of numbers, the probability that the two numbers have no common factor is 6/π², which equals about 0.61.

In other words, any two numbers greater than 1 have about a 61 percent chance of having no common factor (the number 1 doesn't count because it is a factor of any whole number).

For example, if you factor the numbers 22, 98, and 123, you get:

22 = 2 ✕ 11

98 = 2 ✕ 7 ✕ 7

123 = 3 ✕ 41

The numbers 22 and 98 have the number 2 as a common factor. The pair 22 and 123 has no common factor except 1. The pair 98 and 123 also has no common factors.

TRY IT!

A computer's random-number generator created this map showing a hundred stars scattered haphazardly across the sky (above). Use a standard protractor to measure angles, to the nearest degree, formed by various combinations of three stars.

Matthews used the positions of stars in the sky as his source for generating random numbers. He measured the different angles formed by the hundred brightest stars visible in the sky to come up with a million pairs of numbers. He then used a computer to check each pair of numbers for common factors.

Calculating the percentage of number pairs with common factors allowed him to estimate that pi has the value 3.12772. That's within 0.5 percent of the actual value of pi.

In this strange way, Matthews really did find pi in the sky!

NEXT: Galactic Gridlock