## The Normality of PiSeptember 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 forall 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.
## References:- 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.
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. |
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