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Math Science

Pi: Less Random Than We Thought 416

Posted by timothy from the at-least-statistically dept.
Autoversicherung writes "Physicists including Purdue's Ephraim Fischbach have completed a study comparing the 'randomness' in pi to that produced by 30 software random-number generators and one chaos-generating physical machine. After conducting several tests, they have found that while sequences of digits from pi are indeed an acceptable source of randomness -- often an important factor in data encryption and in solving certain physics problems -- pi's digit string does not always produce randomness as effectively as manufactured generators do."
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Pi: Less Random Than We Thought More

Pi: Less Random Than We Thought

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  • This just in: (Score:2, Funny)

    by falzer ( 224563 )
    PI is exactly three.

  • because it ain't random (Score:3, Funny)

    by frovingslosh ( 582462 ) on Sunday May 01, 2005 @02:28PM (#12399478)
    Gee, they found that pi wasn't random. Imagine that. Maybe someday we'll even be able to predict the value of pi.

    • Re:because it ain't random (Score:5, Funny)

      by mobby_6kl ( 668092 ) on Sunday May 01, 2005 @02:41PM (#12399568)
      That's only because they forgot to randomize first!

    • Re:because it ain't random (Score:4, Insightful)

      by i_should_be_working ( 720372 ) on Sunday May 01, 2005 @02:54PM (#12399665)
      It's not that Pi is random or was ever though to be. But you can generate random (or not so random according to the article) numbers by picking out single digits from Pi.

      So I could take, for example, every 14th digit in Pi and that would make a good random string of numbers between 0 and 9.

      • Re:because it ain't random (Score:4, Informative)

        by Anonymous Coward on Sunday May 01, 2005 @03:11PM (#12399813)
        Yes, picking every 14th digit of Pi may share the properties of a good random number (although, once again, the article points out that it is not as good as a RNG). However, actually using said digits for anything would be very unwise, since it wouldn't be that hard to determine that you were using a periodic subsequence of Pi's digits. I.e. don't use this for cryptographic keys.

      • Re:because it ain't random (Score:4, Insightful)

        by Rei ( 128717 ) on Sunday May 01, 2005 @03:19PM (#12399880) Homepage
        I don't see why one should expect Pi to be the ultimate in mathematical random number generation. Its chaos comes from the fact that it is an iterative function; why should we assume that this particular iterative function generates more chaos than others? That would be too convenient.
      • So I could take, for example, every 14th digit in Pi and that would make a good random string of numbers between 0 and 9.

        But if you did it again it wouldn't be as random as a random number generator.

        • Re:because it ain't random (Score:4, Informative)

          by calambrac ( 722059 ) on Sunday May 01, 2005 @03:30PM (#12399978)
          Being able to reproduce random sequences is a good thing. Let's say you want to set up a test that feeds random data into a program until it crashes. It would be nice to be able to rerun that sequence (without having to store the sequence) to make sure the problem gets fixed.

          That's why most random number generators let you specify a seed value. As long as you use the same seed value, you get the same sequence back. If you want a new sequence every time, peg your seed value to some number that varies, like the current time...
    • I an amazed that software custom designed for a task is better at that task than something not!
    • We can already predict the value of pi as accurately as almost anyone would need. I don't know the number of digits that have been computed offhand but I believe it is in the billions. There aren't very many applications that would want or could use more than that, and for most things that is way more accuracy than needed.
      • The computed digits number doesn't even matter anymore since we can calculate the value of a given byte of PI without having calculated any of the previous ones.
        As a side note, the current record by the Kanada lab is 1.2411 trillion digits, their previous record was a bit above 200 billions.
    • "The" value of pi is theoretically unknowable. An approximate value is calculable - many values are calculable. Yet pi does have a more exact value.
      • I don't agree with the term unknowable. Pi is certainly knowable. It just can't be expressed as a finite string of digits after a decimal point. But even if it were unknowable, that doesn't mean it is random. There are many algorithms in mathamatics that produce infinite series, but that doesn't mean they are random. Look at fractals for one example. A very simple math formula can produce an infinite and extremely complex mathmatical result, but even though that result is infinite it is certainly not random

      • Re:because it ain't random (Score:2, Funny)

        by Anonymous Coward
        " "The" value of pi is theoretically unknowable. An approximate value is calculable"

        And drum roll please:

        3 !

  • We're surprised? (Score:2, Interesting)

    by Anonymous Coward

    Uhh.. we're surprised? Pi can be described by numerous simple iterative formulas. When we do that with especially built algorithms we get pseudo random numbers.

    I'd expect pi to be much worse than a PRNG.

  • Computing any digit of pi (Score:5, Interesting)

    by G4from128k ( 686170 ) on Sunday May 01, 2005 @02:33PM (#12399512)
    Given that its possible to compute any digit of pi without computing the preceding digits [sciencenews.org] its not surprising that the digits have structure. The bizarre part of this algorithm is that computes digits in hexadecimal.

  • I had great hope when clicking on the link... (Score:5, Insightful)

    by Anonymous Coward on Sunday May 01, 2005 @02:36PM (#12399540)
    ... but it seems a shitty research, based on the article:

    > Pi never scored less than a B on the tests, and in one case outperformed all the RNGs, which in addition to mathematical algorithms included a device that uses turbulence in a fluid as its source of randomness. But in most cases, pi lost out to at least one RNG, and in several it finished decidedly in the middle of the pack.

    Obviously. There is no reason that pi would beat every RNG out there on a sample of numbers. It should just be slightly ahead the pack (if some RNG are bad), or just in the middle (if all are good).

    > "Our work showed no correlations or patterns in pi's number set - in short, pi is indeed a good source of randomness," Fischbach said. "However, there were times when pi's performance was outdone by the RNGs."

    Well, there is a reason why mathematicians consider that statistics are not a branch of mathematic. And such article are a proof of it.

    pi output on the statistical tests were correct (if they werer not, then it would be an important news, as it would imply correlations). The fact that some other RNG generated "better" output for the (relatively) small sample they used is meaningless.

    • Slightly different view.. (Score:5, Insightful)

      by beldraen ( 94534 ) <chad...montplaisir@@@gmail...com> on Sunday May 01, 2005 @03:33PM (#12400016)
      The real issue with statistics is that people who use them generally do not understand them. I get irritated with people all the time when people "prove" some statement. Statistics shows that a sample of the populace has some correlation within some bound that is likely to be true some percentage of the time. So, the real question is: what was the bound and what percentage of the time was the randomness within that bound. If PI's bound exists outside of the statistical error of the bounds of the other tests then one could say that PI is less random; however, it sounds like they indeed found a few tests where PI "beat" the other tests. In other words, the bound PI was within the statistical error of the other tests, but the computed mean was occasionally better. But, occasionally better is to be expected some percentage of the time. If it is with in that number of times, it is as you say, a meaningless conclusion. Statatics within the bounds of error are completely equal. Probability is math, but it is also just very probable that it is used wrong.
  • No I can't prove it, but there is no proof that Pi was random in the first place. This is just an assumption.

    Of course a source you know to be random will be more random than Pi which is still arguably not random.

    • Re:Not Random (Score:2, Informative)

      by OOGG_THE_CAVEMAN ( 609069 )
      Pseudo-random number generators not "known to be random." They are constructed to pass certain statistical tests with high certainty. Output of these generators, however, NOT RANDOM in strongest sense. Instead, are generated by DETERMINISTIC ALGORITHM.

      Pi's digits also pass certain tests for non-correlation.

      OOGG wish correct additional mistaken assumption of yours:
      Knowledge or proof of FACT does not CHANGE fact. ONLY CHANGE OUR KNOWLEDGE.

      Digits of PI either "random" or "not random" depending on definition

  • Completely Unsurprised (Score:4, Interesting)

    by DumbSwede ( 521261 ) <slashdotbin@hotmail.com> on Sunday May 01, 2005 @02:40PM (#12399565) Homepage Journal
    I actually find this completely unsurprising. PI is completely UNRANDOM. It can be compressed very efficiently with the progression Pi =4 (1-1/3+1/5-1/7+1/9- ...). It is even possible to derive binary or decimal digits of PI in isolation with a formula as well. My point is that since the digits can be represented as a formula, they may completely screw up other functions expecting randomness from them. When they do I would predict there is some deeper connection between the function being tested and Pi than is realized.

    When you cite for example a deviation from a Chi distribution, then there is probably some connection between Chi and Pi that doesn't seem obvious from how Chi is calculated, but is there non-the-less.

    I am not a mathematician (though I did work at Wolfram Research for ten years). I look forward to seeing real mathematicians take on this.

    • Re:Completely Unsurprised (Score:5, Interesting)

      by crmartin ( 98227 ) on Sunday May 01, 2005 @03:15PM (#12399843)
      Well, define "random". The digits of pi occur equiprobably (I believe this is proven) and so represent a random number in the usual sense.

      On the other hand, as you say, they're essentially "pseudorandom" in the sense that they can be computed by a deterministic program.

      What you're groping for here is called "Kolmogarov complexity", or sometimes "Kolmogarov- Solomonov- Chaitin complexity" which can be defined as the length in instructions for some fixed machine of the shortest program that can compute an output sequence. If, without loss of generality, we choose something like a conventional machine, you can think of this as the length of the shortest program in bits.

      What's kind of amazing about it is that there is a supremely elegant and simple proof that there are "really" random sequences in the sense that there is no program that can compute and output a sequence random sequence R that's any shorter than "print R". This is what you're looking for in the sense you're talking about "randomness".

      (The proof comes directly from the fact that there are more bit sequences of length n than there are sequences of length (n-k) for k>0. Thus there must exist sequences of length n which can't be computed by a program of length (n-k).)

      This leads to all sorts of cool stuff, including things like a unification of Gödel's Proof, Turing's Halting proof, Hilbert's Tenth Problem, and chaos theory.

      To learn more, Google for "algorithmic information theory" and "Gregory Chaitin".
  • ...of pi. It's not random at all, I always get 3.14159....

  • Pi experiments and random numbers (Score:4, Interesting)

    by rice_burners_suck ( 243660 ) on Sunday May 01, 2005 @02:44PM (#12399592)
    I would say that it's not too wise to get random digits from pi anyway, because it's too obvious a source. It's also not too difficult, what with storage and whatnot nowadays, to store about ten billion digits and then, knowing a few digits in the sequence, perform a quick pattern match, find it in pi, and know the next digit in the sequence.

    I was wondering, maybe not more than an hour ago, why not get a TV card and gather randomness from there? There are lots of channels on TV, and they have both a video and an audio component. You could set the thing up to change channels at random intervals, and gather things like the color of random pixels at random times, the frequency of random sounds, etc. Perhaps you could use a radio card to do something very similar with the radio. That, combined with entropy from the keyboard, mouse, the time between interrupts of various kinds, the contents of various processor registers or random memory locations, or whatever, should provide basically a random pool that is so random, you'll never have to worry about security problems with relation to them.

    Speaking of which, there are ten digits used in our radix 10 notation; if you want to store a character string in a strange format, you could conceivably store two digits in one byte, because four bits are enough to describe all ten digits, leaving plenty of room for things like a decimal point or a negative sign. I'm saying this because it's not too terribly expensive these days to get a terabyte of storage. If you store, on this terabyte, nothing but digits from pi, in this space-saving format I'm describing, you could store 2,417,851,639,229,258,349,412,352 digits from pi. You'd need some kind of cluster, like PI@home, to compute all those digits. Once computed, who said you can't use pattern-matching algorithms to see if there isn't some kind of pattern? I still believe that somewhere in there, there is a pattern, though it is very large. Hell, who said you can't get an exabyte of storage and do this? If anything, it could become one component in a random number generator that simply never repeats itself.

    • if you want to store a character string in a strange format, you could conceivably store two digits in one byte, because four bits are enough to describe all ten digits, leaving plenty of room for things like a decimal point or a negative sign...

      Wow! What an unusual format. Lets think of a name for it. Hmmm... it's in binary, but it's encoded in a decimal form... I know, lets call it Binary Coded Decimal! It even has a catchy acronym (BCD) that fairly rolls off the tongue. Wow! Maybe we could get so

    • Why bother with channels/channel changing?

      Just dont tune in a channel and listen to the noise/take only the least significant bits.
      Should work more reliable.

  • Did Sagan See This? (Score:4, Funny)

    by NOLAChief ( 646613 ) on Sunday May 01, 2005 @02:44PM (#12399600)
    Calculate it in base 11. Eventually you'll get a sequence of zeroes and ones that when arranged into a square raster form a circle.

    Or so I'm told... :)

  • More on pi and randomness (Score:4, Informative)

    by karvind ( 833059 ) <karvind.gmail@com> on Sunday May 01, 2005 @02:47PM (#12399614) Journal
    The randomness of Pi: Frequency of the digits and Patterns appearing in the number Pi [pcministry.com].

    ScienceNews article (2001) on Randomness of Pi's digits [sciencenews.org]

    Interesting work from Johan on Testing the a-periodic randomness of and comparing it with a Quantum Mechanical source. [versatel.nl]

    But are the digits truely random ? In 1996, NERSC [nersc.gov] Chief Technologist David H. Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found a new formula for pi. This formula permits one to calculate the n-th binary or hexadecimal digits of pi, without having to calculate any of the preceding n-1 digits. This formula was discovered by a computer, using Bailey's implementation of Ferguson's PSLQ algorithm

  • chaos (Score:3, Interesting)

    by potpie ( 706881 ) on Sunday May 01, 2005 @02:48PM (#12399626) Journal
    Fractals, which resemble nature, are not random though they appear to be. Therefore, I've often considered all the universe to be one giant, multi-dimensional fractal.

    I think "random" has a misleading connotation. Just because something is highly unpredictable, it is not necessarily without pattern. We take "random" to mean something that cannot ever be predicted, that follows no pattern. But attractor fractals and many areas of Chaos Theory have proved that there are patterns that defy the human pattern recognition faculty (or at least require the use of a pencil, calculator, super-computer, etc.).

  • Physicists have completed a study comparing the randomness in Darl Mcbride's brainwaves to that produced by 30 typing rats. After conducting several tests, they have found that while sequences of digits from Darl are indeed an acceptable source for randomness, Darl's digit string does not always produce randomness as effectively as rats if the rats are using unixware.

  • Rnadomn (Score:3)

    by EpsCylonB ( 307640 ) <eps&epscylonb,com> on Sunday May 01, 2005 @02:59PM (#12399703) Homepage
    Isn't randomn just something we can't understand ?. Technically speaking if we had enough infomation nothing would be considered randomn. I guess with encryption you pick something thats pretty damn complex and then hope that your competitors agree with you.
    • Isn't randomn just something we can't understand ?

      No, random is when there is a equal chance that either of the elements are picked, and there is no way to determine what was picked afterwards.

    • Re:Rnadomn (Score:3, Insightful)

      by Vellmont ( 569020 )

      Technically speaking if we had enough infomation nothing would be considered random.
      That might be true, except for the heisenburg uncertainty principle. In short it says you can never determine both the exact position of a particle and its momentum. The essential problem is that measurement of either of these properties disturbs the thing you're trying to measure in an unpredictable way.

      The end result is that you can never have enough information. Randomness isn't a lack of understanding, it's a fund
  • I know this has been said before, but perhaps not in this way. Pi is a number that represents a (ideal) physical phenomenon. Yes, it's complex and (probably) infinite, but it still is a numeric representation of an exact property. To me, that automatically presupposes that by its nature it's ordered.

    The only reason anyone could think it would be a good indicator of randomness is because its complexity goes beyond the comprehension of man or machine. I'm not a professional mathematician, so there's not a lot about the nature of pi I can comment on, but it seems to me that in being an ordered number that describes a physical phenomenon, pi has about as much chance to produce randomness as counting the number of leafs on clovers.

    • I am not a statistician, nor have I taken a statistics class, but I can help explain what makes Pi "random" in the sense they're talking about.

      In lower level statistics courses, to pick random numbers, you use a line from a random number table. These numbers were generated using some pseudo-random methodology like observing a process that's nearly random.

      Now, Pi comes into this because it's an irrational, so it has a non-repeating decimal expansion. You can use that decimal expansion to cherry pick number
  • 24 is the highest number, you bunch of fazools. Fuhgeddaboutit [tvtome.com].
  • At the risk of losing karma, what an obvious statement!! Pi is a mathematical number used to calculate certain things such as circumference OF COURSE ITS NOT RANDOM, if it was then we wouldn't be using it for so many important functions
  • It's the same every time.

    Duh.
  • and here I was using MS Solitaire to generate my random numbers. :P

    Does the concept of anything "random" really exist or is it just a word synonymous with obfuscation?

  • This has bothered me since I first ran across it in a colloquium when I was in grad school (in math) in the early '90s. It's more a matter of symantics than anything else, but it still bugs me becuase, like the difference between the words "secure" and "vulnerable", it leads to a lot of confusion.

    Let's say I ask you if any old sequence of numbers (like this one) is random or not:

    1,0,1,0,1,0,....

    The correct answer is that you can't tell me unless I also tell you HOW the sequence was generated: Did

  • In school (Score:5, Funny)

    by gsasha ( 550394 ) on Sunday May 01, 2005 @03:33PM (#12400014) Homepage
    I had a teacher who insisted that Pi is exactly 3.14, and that the radiation after nuclear explosion decays by a factor of 2 in exactly 5 hours.
    Admittedly, he wasn't a math teacher though...
  • On my computer, the article title shows as "A study on the randomness of the digits of ¼" (that's 1/4 if the character shows incorrectly on your desktop).

    It's a Firefox 1.0.3 on a Mandrake 10.2.

  • From a mathematician ... (Score:3, Insightful)

    by rkmath ( 26375 ) on Sunday May 01, 2005 @04:17PM (#12400378)
    (1) "Pi is not random becuase I have a formula for its digits" is nonsense. Randomness is not the inability (or impossibility) to predict (at least in this situation). Randomness refers to statistical properties of the sequences. For ex. no correlation between conseq. digits, no corr. betweteen conseq. pairs of digits and so on brings a sequence closer to randomness.

    (2) If you REALLY want randomness (with impossibility of prediction, and unreplicability of the sequence) - go and count events in a radiactive decay experiment. (More precisely, count waiting times for each successive decay - they follow an exponential distribution). (I think fourmilab has a 1-time rnadom number generator linked up to a geiger counter - don;t remmeber the URL any more).

    (3) Why do mathematicians find "randomness" in digits interetsing? The reasons are similar to why people prove theorems about "how randomly are the primes distributed among the integers". It says something about the structure of the primes. I am not a number theorist - so I cannot give explicit results.

  • Al-Kashi, a cool mathematician (Score:5, Interesting)

    by kronocide ( 209440 ) on Sunday May 01, 2005 @04:18PM (#12400385) Homepage Journal
    I must tell you a story.

    In the first half of the 15th century the Persian mathematician Al-Kashi calculated pi to 14 places. It would be over a hundred years until a European calculated it to 9 places. But that's not what makes Al-Kashi cool, the Arabs where so much better at math in that period. What made him cool was that he stopped. He observed that, with his pi, the calculation of the circumference of a circle with a radius twice the size of Earth would have a margin of error smaller than a "horse hair" (a Persian unit). Problem solved, next problem. Meanwhile, people are still today using computers to get pi to _hundreds_of_billions_of_decimal_places!! As if there's something unique about pi because it's irrational and transcendental, when this is in fact true of the vast majority of all real numbers. Here's to Al-Kashi, a sane man and a pragmatic!

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