## Pi DayMarch 14, 2012 Today is "Pi Day" (3.14). I wasn't among the many fans who likely stayed awake until 1:59 AM to get those last extra decimal digits, but I'm still a big fan of the mathematical constant, pi. Pi appears to be an important part of the physical universe, since it appears in so many fundamental equations of physics. For example, the Einstein field equations of general relativity contain the constant (8πG/c^{4}).
Pi must have weighed heavily on my mind in 2010, since I wrote three articles about it.
• Buffon's Needle, July 19, 2010The last article somehow made it to the fourth page of 43,400 results on Google when I searched for hex digits of pi. It must be a niche area. More on hex digits of pi later in this article.
The first accurate estimates of pi were done by Archimedes, who used a geometrical approach of approximating a circle by inscribed and circumscribed polygons (see figure). You can do this for regular polygons of any number of sides, but Archimedes was limited to those with special angles, such as 30 degrees, for which he could calculate the tangent.
Today, using trigonometric tables, we can do this for any regular polygon. This doesn't make that much sense, since the trigonometric functions are calculated by series expansions, and so is pi.
P of the polygons and ratio them to the radius R of the circle to get
π ≈ P/(2R),Archimedes was able to calculate our elementary school value of pi, 22/7, using a circumscribed 96-sided polygon. Ptolomey did the same with a 360-sided polygon, getting the value, 3.141666..., that's correct to three decimal places. Computer techniques have allowed a calculation of pi to 10,000,000,000,050 decimal digits. This was done by Shigeru Kondo, a Japanese systems engineer, on home-built hardware. This is truly overkill. Our present estimate of the diameter of the universe is 93 billion light-years (8.8 x 10 ^{26} meters), so we only need twenty-seven digits of pi to calculate the circumference of the universe to better than a meter, and thirty-eight digits to get a value of the circumference closer than the size of an atom.
In the past, if you wanted the n-th digit of pi, you needed to calculate all the digits leading up to n. This changed in 1997, when David Bailey, Peter Borwein and Simon Plouffe (BBP) published a formula that allows calculation of any digit of pi without calculation of all the preceding digits.[1-3]
^{k} term probably tipped you off, the equation gives pi in hexadecimal form.
I've posted C source code for calculation of the first 20,000 hex digits of pi in a previous article, but it's worth a few extra electrons to post the source code of an improved version. It's based on code published by David Bailey,[4] and it will calculate these 20,000 hex digits in under a minute on most computers.
Mathematician, Clifford Pickover, published an interesting conjecture in May, 2003.[5] This conjecture, which depends on the digits of pi being truly random,[6] is as follows:
"Somewhere inside the digits of pi is a representation for all of us -- the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal."You can try to find parts of yourself in pi, at this web site.[7] ## References:- David H. Bailey, Peter B. Borwein and Simon Plouffe,"On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation, vol. 66, no. 218 (April 1997), pp. 903-913.
- David H. Bailey, "The BBP Algorithm for Pi, September 17, 2006
- Tsz-Wo Sze, "The Two Quadrillionth Bit of Pi is 0! Distributed Computation of Pi with Apache Hadoop," arXiv Preprint Server, August 18, 2010.
- David H. Bailey, "BBP Code Directory" (Updated September 8, 2006)
- Cliff Pickover, "We are in Digits of Pi and Live Forever," May 2003.
- David Whitehouse, "How random is pi?," BBC News, July 23, 2002.
- David G. Andersen, The Pi Searcher Web Site
- David Blatner, Joy of Pi Web Site.
- Eric W. Weisstein, "Pi Digits." From MathWorld--A Wolfram Web Resource.
Linked Keywords: Pi Day; decimal digits; mathematical constant, pi; physical universe; fundamental equations; physics; Einstein field equations; general relativity; 2010; Buffon's Needle, July 19, 2010; Another Piece of Pi, July 28, 2010; Pi is Forever, September 23, 2010; Google; hex digit; Archimedes; geometry; polygon; regular polygon; tangent; trigonometric tables; series expansion; Inkscape; perimeter; radius; elementary school; Ptolomey; 10,000,000,000,050 decimal digits; Japanese; diameter of the universe; light-year; meter; size of an atom; David Bailey; Peter Borwein; Simon Plouffe; formula; C source code; electron; Clifford Pickover; conjecture; random. |
RSS Feed
## Google Search
Latest Books by Dev Gualtieri
- Refining Germanium - June 22, 2017
- Granular Capillarity - June 19, 2017
- Kirchhoff–Plateau Problem - June 15, 2017
- Self-Assembly - June 12, 2017
- Physics, Math, and Sociology - June 8, 2017
- Graphene from Ethylene - June 5, 2017
- Crystal Alignment Forces - June 1, 2017
- Martian Brickwork - May 29, 2017
- Carbon Nanotube Textile - May 25, 2017
- The Scent of Books - May 22, 2017
- Patterns from Randomness - May 18, 2017
- Terpene - May 15, 2017
- The Physics of Inequality - May 11, 2017
- Asteroid 2015 BZ509 - May 8, 2017
- Fuzzy Fibers - May 4, 2017
- The Sofa Problem - May 1, 2017
- The Wisdom of Composite Crowds - April 27, 2017
- J. Robert Oppenheimer and Black Holes - April 24, 2017
- Modeling Leaf Mass - April 20, 2017
- Easter, Chicks and Eggs - April 13, 2017
- You, Robot - April 10, 2017
- Collisions - April 6, 2017
- Eugene Garfield (1925-2017) - April 3, 2017
- Old Fossils - March 30, 2017
- Levitation - March 27, 2017
- 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
### Deep ArchiveDeep Archive 2006-2008
Blog Article Directory on a Single Page |

Copyright © 2017 Tikalon LLC, All Rights Reserved.

Last Update: 06-22-2017