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. 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.
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.
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.
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.
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.
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 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.)
- David Richeson, "Circular reasoning: who first proved that C/d is a constant?" arXiv, March 14, 2013.
- Carlos Sevcik, "Fractal analysis of π normality," arXiv, July 28, 2016.
Permanent Link to this article
Linked Keywords: Ratio; circle; circumference; diameter; mathematics; mathematical; pi; Greek; mathematician; Archimedes (287-212 BC); constant; approximation; estimate; scholarly method; academic publishing; paper; fiber; string; length; ruler; experiment; hexagon; perimeter; polygon; numerical digit; computer program; rounding; single-precision floating-point format; single-precision (float); C (programming language); Archimedes (287-212 BC); approximation; estimate; pi; Wikimedia Commons; Inkscape; infinity; irrational number; square-root of two; mathematical proof; legend; deity; god; Hippasus (c. 450 BC); drowning; drown; ocean; sea; sacrilege; unholy; randomness; random; conjecture; normal number; radix; number base; decimal; base 10; Blowfish cipher; hexadecimal; statistics; statistical; probability distribution; truth; mathematical constant, e; natural logarithm; arXiv; Carlos Sevcik; Venezuelan Institute for Scientific Research; Instituto Venezolano de Investigaciones Científicas (IVIC, Caracas, Venezuela); fractal analysis; uniform distribution; convergent; converge; fractal; Benoit Mandelbrot (1924-2010); École Polytechnique Fédérale de Lausanne; Rama; waveform; Chudnovsky algorithm; Ramanujan series; Chudnovsky brothers; Linux; Mersenne Twister (MT19937); random number generator; Diehard tests; DIEHARD; Cartesian coordinate system; graph; fractal dimension; Gnumeric.
Latest Books by Dev Gualtieri
Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!
Blog Article Directory on a Single Page
- Soybean Graphene - March 23, 2017
- Income Inequality and Geometrical Frustration - March 20, 2017
- Wireless Power - March 16, 2017
- Trilobite Sex - March 13, 2017
- Freezing, Outside-In - March 9, 2017
- Ammonia Synthesis - March 6, 2017
- High Altitude Radiation - March 2, 2017
- C.N. Yang - February 27, 2017
- VOC Detection with Nanocrystals - February 23, 2017
- Molecular Fountains - February 20, 2017
- Jet Lag - February 16, 2017
- Highly Flexible Conductors - February 13, 2017
- Graphene Friction - February 9, 2017
- Dynamic Range - February 6, 2017
- Robert Boyle's To-Do List for Science - February 2, 2017
- Nanowire Ink - January 30, 2017
- Random Triangles - January 26, 2017
- Torricelli's law - January 23, 2017
- Magnetic Memory - January 19, 2017
- Graphene Putty - January 16, 2017
- Seahorse Genome - January 12, 2017
- Infinite c - January 9, 2017
- 150 Years of Transatlantic Telegraphy - January 5, 2017
- Cold Work on the Nanoscale - January 2, 2017
- Holidays 2016 - December 22, 2016
- Ballistics - December 19, 2016
- Salted Frogs - December 15, 2016
- Negative Thermal Expansion - December 12, 2016
- Verbal Cues and Stereotypes - December 8, 2016
- Capacitance Sensing - December 5, 2016
- Gallium Nitride Tribology - December 1, 2016
- Lunar Origin - November 27, 2016
- Pumpkin Propagation - November 24, 2016
- Math Anxiety - November 21, 2016
- Borophene - November 17, 2016
- Forced Innovation - November 14, 2016
- Combating Glare - November 10, 2016
- Solar Tilt and Planet Nine - November 7, 2016
- The Proton Size Problem - November 3, 2016
- Coffee Acoustics and Espresso Foam - October 31, 2016
- SnIP - An Inorganic Double Helix - October 27, 2016
- Seymour Papert (1928-2016) - October 24, 2016
- Mapping the Milky Way - October 20, 2016
- Electromagnetic Shielding - October 17, 2016
- The Lunacy of the Cows - October 13, 2016
- Random Coprimes and Pi - October 10, 2016
- James Cronin (1931-2016) - October 6, 2016
- The Ubiquitous Helix - October 3, 2016
- The Five-Second Rule - September 29, 2016
- Resistor Networks - September 26, 2016
- Brown Dwarfs - September 22, 2016
- Intrusion Rheology - September 19, 2016
- Falsifiability - September 15, 2016
- Fifth Force - September 12, 2016
- Renal Crystal Growth - September 8, 2016
- The Normality of Pi - September 5, 2016
- Metering Electrical Power - September 1, 2016
Deep Archive 2006-2008