The search for pi


Today March 14 is so-called ‘pi day’ where in the month/day format in the US for the date gives 3/14. The number pi (the ratio of the circumference to the diameter of a circle) is a source of endless fascination because it is so basic and yet so ubiquitous in all areas of science and mathematics and everyday life. Here is a fascinating article by Xiaojing Ye on the search for obtaining increasingly precise values of pi. It contained many nuggets of information that were quite new to me.

Various techniques have been used to calculate pi. The earliest written approximations were obtained between 3,000 and 4,000 BC in Babylon and Egypt using trial-and-error methods and they obtained values close to 3.1. The next level of precision was reached by a technique developed by Archimedes around 250 BC based on geometrical approximations using circles with inscribed and circumscribed regular polygons to set lower and upper limits and he got a value of 3.14. Ptolemy extended this method in 150 AD to get a value of 3.1416.

A Chinese mathematician named Liu Hui in 265 AD used a polygonal algorithmic technique to also get four digits but that method was extended by Zu Chongzhi in 480 AD to get seven digits. That record held for 800 years. Austrian mathematician Christoph Grienberger in 1630 used that same technique to get 38 digits “which is the most accurate approximation manually achieved using polygonal algorithms.”

The next breakthroughs came with the development of infinite series in the 16th and 17th centuries.

In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 – the best approximation achieved without the aid of a calculator or computer.

The development of iterative algorithms for calculating pi, combined with the arrival of computers, saw an explosion in the number of digits achievable.

Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!

We are well beyond the level of accuracy necessary for purely practical purposes and the search now serves other needs.

It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.

What I also found interesting were the theorems that proved things about what pi was not.

Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number – it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (such as pi2=10 or 9pi4-240pi2+1492=0).

Fascinating stuff.

Update: A friend of mine informs me of the Bailey–Borwein–Plouffe formula which is an algorithm that enables you to directly calculate any given binary digit of pi without having to calculate any of the preceding digits. The catch is that the further out the digit is in the sequence, the longer it takes to calculate it.

Comments

  1. Paul Neubauer says

    In fact, not only is today pi day, but it’s a special pi day in that this is the best approximation we are ever going to get with a two-digit year at 3/14/16. We have to go back more than 400 years to get a better approximation from the date of 3/14/1593.

  2. speedwell says

    A common question is whether, if the digits of the decimal representation go on forever and never repeat, there is an example of every possible sequence of numbers somewhere in the representation, if you go far enough “down” into it. The answer is no, and the proof is trivial, but it’s a satisfying hunt to find out why it’s so.

  3. speedwell says

    Well, I should correct myself. The answer is really “a nonrepeating infinite string must not necessarily contain every possible sequence”.

  4. says

    According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter.

    So it’s more than sufficient for the Large Hadron Collider, with it’s 27km circumference. I would have expected CERN and others involved to be so anal retentive in their precision that they built it to thousands of decimal places. Especially with the amount of money involved.

  5. Holms says

    This is true for any nation using M-D-Y, i.e. America, Belize, Micronesia, Phillipines, Saudi Arabia, and Canada. In the entire rest of the world, not so much. Oh and while I was looking that up, I see Canada is listed as using all three formats… that must get confusing.

  6. anat says

    Continuing speedwell’s question – does anyone remember the ending of Carl Sagan’s ‘Contact’? In the book version, it is discovered that deep into the digit’s of pi, in some base that is not 10 (but I don’t remember which one) there is a sequence whose length is a square of a prime number, composed entirely of 2 digits, that forms the image of a circle if displayed in lines the length of said prime number. Would such a finding be something that should be expected, given the infinity of pi, or would it be reasonable to suspect it was a message from the creator of the universe?

    And if we are here anyway, is it possible to have a universe in which pi has some different value?

  7. Richard Simons says

    I see Canada is listed as using all three formats… that must get confusing.

    It does. I usually spell out the month.

  8. Rob Grigjanis says

    left0ver1under @4: High-energy physics experiments aren’t high-precision. Measurements and theoretical values are typically in the few-percent accuracy range. It’s the low-energy stuff, like anomalous magnetic moment, that can achieve one in 10^9 or better accuracy.

  9. Mobius says

    Just a little mathematical trivia…

    Numbers which are solutions to rational algebraic equations (that is, polynomials with rational coefficients) are called “algebraic”. (Imagine that.) Numbers which are not are called “transcendental”. Pi and e are examples of transcendental numbers. Actually, the algebraic numbers are countable, which means the vast majority of real numbers are transcendental, which are uncountable. (Countable and uncountable are actual mathematical terms describing the cardinality of infinite sets.)

  10. Mobius says

    @7 Richard Simons

    Even after so many years, I still use the military date system which gives d-m-y with the month as a 3 letter form and the year the last 2 digits. For example, today is 14Mar16. The system always seemed so logical to me, and clear to understand.

  11. says

    Mobius (#10) –

    14Mar16. The system always seemed so logical to me, and clear to understand.

    If you find that “logical”, where did you buy your watch that displays time as seconds-minute-hours?

    Naming things YYYY-MM-DD not only makes more sense than any other, computers automatically sort files and directories which are named that way. It makes finding stuff a lot easier.

  12. Rob Grigjanis says

    Henry Gale @12: It’s obvious to anyone with a lick of sense that the proper circle constant is 2τ = 4π, since that is the number of radians required to rotate a spinor back to its initial value. The celebration date is Jan 25th, and it should be called ω, because it ends the discussion!

  13. Holms says

    And if we are here anyway, is it possible to have a universe in which pi has some different value?

    Yes! All you need is a universe in which space is not flat.

  14. hyphenman says

    @No. 11 left0ver1under

    I’ve used a pseudo-star date system—YYMMDD.TIME. I’m writing this at 160315.0328— for years for just that reason.

    Jeff Hess

  15. hyphenman says

    Hmmmm… I’ve never noticed before but I posted No. 16 at 0328, but the FTB server logged the event at 0337.

    This is posting at 0334

  16. Marshall says

    @speedwell #2: The property you’re looking for is normality: it’s a number where the distribution of digits is uniform, regardless of the base.

    Pi has not been proven to be normal, but it is suspected to be so.

  17. Marshall says

    To continue my last comment: if a number is normal, then it does indeed contain every possible sequence, in which case–yes, Shakespeare is somewhere in pi (in any possible encoding scheme).

    To go even further, every possible model simulation of our universe, and all other universes, is also encoded somewhere in pi. But…pi’s not special; the same can be said of any normal number.

  18. Rob Grigjanis says

    Holms @15:

    All you need is a universe in which space is not flat.

    But, if you’re defining it as circumference/diameter of a circle (and if you can unambiguously define a circle), pi wouldn’t be constant. It could depend on location, orientation and would certainly depend on size.

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