### Random Pi

My Uncle Walter was an interesting character. He served as a Marine in World War II, rolled his own cigarettes, and was into recycling before it became popular. He was high school educated only, as were most of his generation, and when he told people what his nephew did, he described it as "Pi-R-Squared-Physics." He realized in his own way that the mathematical constant, pi, has a special place in physical reality. I was reminded of this when I read a recent news article about a crop circle in Britain that encoded the decimal value of pi to high precision [1]. A crop circle is a geometrical pattern created in an agricultural field by flattening the crop. Although some people believe that these are messages from extraterrestrials, they are generally considered to be school boy pranks, although these must be well educated school boys.

I've discussed pi in several previous articles [2-5]. Pi is a transcendental, irrational number with a number sequence that does not repeat. Pi is part of many fundamental equations of physics, which indicates to me that pi has a physical reality independent of an observer. Pi is considered to be so important to mathematics that it's been calculated to more than a trillion digits. This is definitely overkill for most applications, since only fifty digits of pi are needed to calculate the circumference of the observable universe to within a proton's width. The first few decimal digits of pi are

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

A casual glance indicates quite a few more threes and nines after the decimal point than randomness would indicate. There are eight threes and eight nines, but only two zeros. This is only a small statistical sample, so it takes many more digits to discover that pi is indeed random. Interested readers can find listings of many digits of pi on the internet [6] and one approach to random number testing [7].

Testing pi for randomness is an interesting experiment, the results
agree with our expectations, but there's no theory that states this
must be the case. In fact, we should admit that pi is *not* random, since it can be calculated. Something that can be calculated is, *a priori*, not random. This was expressed succinctly by computer pioneer, John von Neumann,
when he said, "Any one who considers arithmetical methods of producing
random digits is, of course, in a state of sin. For, as has been
pointed out several times, there is no such thing as a random number,
there are only methods to produce random numbers; and a strict
arithmetic procedure, of course, is not such a method."

Von Neumann was interested in randomness, since he participated in the development of the Monte Carlo method. To speed his computer calculations, von Neumann developed a simple random number generator, called the middle-square method. This generator has its problems, the most important of which is the limit on the number of random numbers you can get from it (its "period"), but it was useful for the small-scale computer models of his time. It works as follows:

• Step 1: Select a ten-digit number.

• Step 2: Square the number.

• Step 3: Select the middle ten digits of the square

• Step 4: This is your random number

• Step 5: Use this as your new ten-digit number and go to Step 2

If you use n-digit numbers, this procedure will give you 10^{n} random numbers, although there are a few cases, involving too many zeros in the middle numbers, in which it fails.

John von Neumann was born on December 28, 1903. The sequence 122803 occurs starting at position 1,600,569 in the digits of pi after the decimal point [8]. The number 12281903 does not occur in the first 200,000,000 digits of pi.

References:

1. Baffling crop circles equal pi.

2. Mathematical Objects (This Blog, May 9, 2008).

3. Three Hundred Articles, (This Blog, November 13, 2007).

4. Happy Birthday Albert ! (This Blog, March 14, 2007).

5. 100,000 Digits of Pi (This Blog, October 10, 2006).

6. Pi Pages on the Web (Pi Pages).

7. S.J.
Tu and E. Fischbach, "Geometric Random Inner Products: A Family of
Tests for Random Number Generators," Phys. Rev. vol. E67 (2003), 016113

8. Pi Sequence Finder.

9. Official Web Site for Pi Day.

10. Computing Pi (Wikipedia).

11. Pi Formulas (Mathworld).