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The Normality of Pi

September 5, 2016

The ratio of a circle's circumference to its diameter, a mathematical quantity known as pi (Greek letter, π), or as Archimedes' constant, is valid for any circle. The name of the Greek mathematician, Archimedes (287-212 BC), is attached to this constant, since he was the first to give a close numerical estimate of its value. A scholarly paper also credits Archimedes as the first the understand that the ratio of the circumference of a circle to its diameter is a constant that's the same for all circles.[1] This fact, in itself, has a significance that's rarely discussed.

You can, of course, thread a string around a circle, and then measure the length of the string and the circle's diameter with a ruler to get a value of pi. That, however, is an experiment, and it's not something a proper mathematician does. I presented several experimental methods for the determination of pi in two earlier articles (Buffon's Needle, July 19, 2010, and Another Piece of Pi, July 28, 2010.

Archimedes placed hexagons within and around a circle, and he reasoned that pi has a value between the perimeters of the inner and outer hexagon. Not stopping there, Archimedes proceeded to 96-sided polygons, showing that 223/71 ≤ π ≤ 22/7; that is, 3.1408 ≤ π ≤ 3.1429. If we average these, we get 3.141855, which is within a hundredth of a percent of the value of pi. I use just the first eleven digits of pi, 3.1415926535, in most of my computer programs. That last 5 should be rounded up to 6, but I don't bother. A single-precision (float) in C is precise to about 7 digits.

Archimedes and circle hexagons

Archimedes (287-212 BC) and his method of hexagons for an estimate of the value of pi. The inner hexagon gives a lower limit of 3.0, while the outer hexagon sums to 3.4641... (Left image via Wikimedia Commons. Right image created using Inkscape)


I stop at eleven digits, but pi has many more; in fact, an infinity of digits. That's because pi is an irrational number, just like the square-root of two. I wrote about irrational numbers and the square-root of two in a recent article (The Square Root of Two, June 6, 2016). The square-root of two was the first number proven to be irrational, and there's a legend that the gods caused Hippasus (c. 450 BC), the discoverer of this fact, to drown at sea for this unholy revelation.

Examination of any segment of the digits of pi suggests that its digits are random. A closer look using computer techniques reaffirms this idea. It's conjectured that pi is a normal number; that is, an irrational number whose digits occur with the same likelihood. This is true in every number base, not just our commonly used base 10. If pi were normal, then none of its digit, or any combination of digits, occurs more frequently than any other. Digits of pi are used as random numbers in the Blowfish cipher.

Spirals of the first few digits of pi in the decimal and hexadecimal representations of pi.

The first few decimal (left) and hexadecimal (right) digits of pi. Presently, about 10 trillion (1013) decimal digits of pi are known. (Created using Inkscape.)


While it's possible to do improved statistical tests of the distribution of the digits of pi, since more and more of its digits are being amassed, there's no real proof of pi's normality. Not only have people looked at the distribution of the digits, themselves, but they've looked at the distribution of pairs of digits (dyads), groups of three digits (triads), up to larger n-ads. All these tests indicate a normal number. While it's almost certainly true that pi and its irrational cousins, √2 and e (the base of natural logarithms) are normal numbers, no mathematician has actually proved this.

A recent paper on arXiv by Carlos Sevcik of the Instituto Venezolano de Investigaciones Científicas (IVIC, Caracas, Venezuela) has added to the evidence of pi's normality, not with an actual proof, but with a different type of statistical test. Sevcik did a fractal analysis of pi's digits, and he showed that it gave the same result as that for a uniformly distributed random succession of independent decimal digits. Sevcik also detected that this measure of randomness for pi converged on the ideal result as more digits of pi were included.[2]

Benoit Mandelbrot (1924-2010)

The father of fractals, Benoit Mandelbrot (1924-2010), at the École Polytechnique Fédérale de Lausanne, March, 2007.

The word "fractal," originated with him.

(Wikimedia Commons image by Rama.)


The fractal analysis would detect whether there are complex structures that exist in the sequence of pi's digits, since it looks at the sequence as a whole, and not just its parts. In this analysis, the digits of pi are considered to be periodic samples of a waveform. For this analysis, Sevcik generated his own list of digits of pi using a variant of the Ramanujan series developed by the famed Chudnovsky brothers. To illustrate how far computing has progressed, this operation took just 1929 seconds on his Linux computer.[2]

If you want to compare the sequence of pi digits to random numbers, you need a good sequence of random numbers, which is not a trivial undertaking. Sevcik used the Mersenne_Twister (MT19937), a random number generator that's passed the DIEHARD suite of statistical tests.[2]

While Sevcik's result is just another statistical argument for the normality of pi, his results, as shown in the graph, are convincing. His fractal analysis demonstrates that the digits of pi and a sequence of random numbers show the same fractal behavior.

Fractal dimension of pi

Fractal dimension of the digits of pi (circles) compared with that of a sequence of random integers (triangles) for sequences up to a billion (109).

(Graphed using Gnumeric from data in tables 1-2 of ref. 2.[2]


References:

  1. David Richeson, "Circular reasoning: who first proved that C/d is a constant?" arXiv, March 14, 2013.
  2. Carlos Sevcik, "Fractal analysis of π normality," arXiv, July 28, 2016.

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