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Share The Pi!
freedumb writes "From this article in Nature: "Two mathematicians have now taken the first step towards proving that pi contains not a single message but every conceivable message, meaningful or not."" Actually, it's a discussion concerning whether "that all strings of the same length appear in pi with the same frequency: 87,435 appears as often as 30,752, and 451 as often as 862, a property known as normality."
Does anyone have working source code for the algorithm to calculate a digit of pi with out knowing the previous digits ? Is this something that my laptop can handle ?
...some guy found 424242 in Pi at position 242424 (counting the the decimal point.)
But, but, but... that means pi contains the Code-Red worm code! (In binary and source form!) Those bastards have infected !
Other irrational numbers obviously do not contain every possible sequence, like 1.01001000100001.. Although there are random irrational numbers that do. Pi is somewhere in between, more complex but obviously not random. (There is no way of talking about a specific random irrational number.) It can be described as the sum of many infinite sequences, some of which are actually rather simple (something like pi/4 = 1-1/3+1/5-1/7.., as I remember). The digits appear "random", but because the series is simple to write out, it's strange that some digits do not appear slightly more than others, but it will be interesting if they can prove it one way or the other.
the size of this "position number" would probably be many many times the size of the text you are trying to compress - i.e., useless.
Using the pi-search engine:
String Length Offset Length
1.......1.......1.......1
11......2.......94......2
111.....3.......153.....3
1111....4.......12700...5
11111...5.......32788...5
111111..6.......255945..6
1111111.7.......4657555.7
so the index has the identical problem...
You know what's even weirder? I can take any number and make it disappear. I think its in my third grade notes, lemme check, oh wait -- here it is.
You can make Pi Disappear by doing:
(PI)*0+(100-(50*2)) = 0 !!! WTF? Is this magic
Lets try it with e,
(e)*0+(100-(50*2)) = 0 !!! Wow, that's impressive.
Ok, now let x = 118903579039072348989123.345325235325323523
(x)*0+(100-(50*2)) = 0 !!! Hmm, this is a dangerous mathmatical formula, what if the Commies figured out they could multiple America by 0 and the Good Ol' USA would disappear.
On a serious note, guys, we've got to keep
this secret, if word got out, people would
multiple any number by 0, secretaries could
make stock worthless by multiplying it by zero, bugger-flipping freaks could make planes crash by multiplying their altitude to zero and people could make Dubya's IQ rise by multiplying it by zero!
~~Cannis
http://www.telalink.net/~mccann
(Yes, I voted for George W. Bush, but only
because Al Gore's a stupid luser and the
people who vote for Nader scare me.)
If all strings are in there, then Pi is illegal, as it contains the DeCSS source code in there somewhere, and therefore can be used in a way contrary to the DMCA. Start locking up anyone using pi now, before it's too late!
I believe that in Indiana Pi was officially 4 at one point. =P
I think it works like this. They used a wheel with a diameter like 1/1000th the height of the pyramid. They then created a square with each side 250 circumfrences of that wheel. They didn't actually need to know PI to create the wheel but the length of the pyramid is now PI times the height.
You mean that position 242424 isn't random, that it was chosen by some higher power to be in that spot?
An amazing finding...
...next thing you know, they'll be finding secret messages hidden in the Bible
using some sort of letter skipping technique.
What is even more interesting is that NetBSD is a subset of this
"The interesting question is, must it also have all infinite strings embedded in it? I suspect that would lead to a contradiction, but this goes beyond my mathematical competence."
Sure would. Take pi, and add 1. Remove the decimal point so that you just have the infinite string 414159265358979... Suppose that's embedded in pi somewhere. The string 414159... must therefore also be embedded in itself (that is, aside from the trivial embedding, where you just look at the entire string). But wait! That means that the string 414159... must repeat, and thus, so must pi. Since pi is irrational, we have a contradiction.
(Sorry for the rather unclear explanation, it's 4 A.M. here and I can barely think straight.)
You should of just gone to google, the first instance I easily found of that story is here. Its worth a quick read.
The point is, nobody ever has proven that Pi contains every possible string of numbers. I'm not even sure it has been proven that all the digits are equi-probable.
OG.
Soo.. if pi contained every possible message (ie, was truly random), couldn't you in theory find a specific position where pi prints out say, the Max Payne ISO, and distribute that position to friends?
Then, said friends, start calculating pi from that offset (wasn't there a story on slashdot about calculating any N digit of pi without having to calculate the first N-1 digits). Voila, kickass compression.
Of course, the small snags here are:
But once you get over these boring details, pi-based-compression can make for some very neat applications
It is more random, however it is not useful for most things as it is very predictable. Think of it as if you are using rand but everyone is using the same seed.
Erlang Developer and podcaster
Really frightening is that Goatsex is encoded somewhere in Pi. An infinite number of times!
--
"How many six year olds does it take to design software?"
dinner: it's what's for beer
> If we could just get enough of the message contained in Pi, maybe some order would magically appear.
I'm sure once it's calculated to an infinite number of digits, the true meaning will become clear.
You must have missed the story where they zipped DeCSS code and came out with a long prime number. The algorithm to get DeCSS code from this number is unzip (and it's available everywhere). So if the number itself is not illegal, then neither is the decimal displacement from pi.
___
___
If you think big enough, you'll never have to do it.
This reminds me of my theory of computation class. A turing machine is arbitrarily long, but not infinitely long. Similarly, any arbritrarily long sub-sequence of e (or even pi) may be found in pi (assuming normality) but not the infinitely long entire sequence of e.
Did you mount a military-grade, variable-focus MASER on an unlicensed artificial intelligence?
No it won't. First, the behaviour of rand() is implementation-dependent. Second, most implementations produce all their possible outputs (typically 2**16) in a fixed cyclic order.
I was thinking the same thing... Can we find the gzipped source of decss in pi and then state the number at which the source code begins and the length? Then all the decompressor would need to do is calucalate arbitrary digits of pi and starting a given number and for a given length. (there's already algorithms for calulating stand alone digits of pi at a given place) And voila! This may have already been said, but I'm lazy, and there's too many comments to read on /. for me to check, I just read this thread. :)
---
For those of you who feel like taking a shot at generating some military secrets or just next year's Xbox hit title, calls to rand() is way faster than generating pi digits. It will come up with the same data as well (every message, that is), in a little more than two shakes of a lambs tail.
Make me think about this article in a French science magazine about "La bibliothèque de Babel". Sorry, don't remember the title.
translation : "The Babel library"
Named after the guy who "invented" it.
This virtual library contains _many_ books...but a finite number of books.
The books contains 60 chars per line, 30 lines per page, and 500 pages per book...so 90000 chars total. (not sure about the numbers, but it's not important)
If the books uses an alphabet of 26 chars (a to z...sorry no caps), the space (" ") and the period char (".") we have 27 possible chars.
So 27^90000 different books (possibilities).
A book with all "a"s. The same book (all "a"s) but with one "b" at the end, a book about your life, a book about your life minus 20 years, a book on how the universe was created, etc... _And_ you have the same version in many languages (all the languages that uses this alphabet). In short, and infinite number of story...hrm wait a minute...it's a finite number (27^90000).
P.S. Sorry for my english
If you think about it, the digits of pi are not really a "random sequence" at all, at least according to Gregory J. Chiatin's theory of algorithmic information theory. The digits of Pi are of course compressible. You can write a computer program which is of finite size that will generate the digits of Pi, and that's definitely smaller than all the digits! The "randomness" only arises from our choice of base, actually. If you would use a factorial base representation (for instance) to write Pi, it wouldn't look very random...
Qu'on me donne six lignes écrites de la main du plus honnête homme, j'y trouverai de quoi le faire pendre.
It's an irrational number. It doesn't end. Nothing more to see, go back to your homes.
If you have an infinate list of numbers, of course you can pull whatever you want out of them. Eventually something interesting will come up. I'm sure places 495865749584 to 495857498745 in binary are linux kernel 6.2.4 compiled perfectly for my hardware. In some other goofy place is DeCSS, and in another is any goofy message you want to look into it.
Not being a troll, but I still don't see the big deal about one irrational number.
No point other than karma whoring off of lame jokes about DeCSS (repeated at least 50 times per Pi story), no.
The answer they should have given you was this: The definition of factorial that we gave you was for your ease of understanding. The real definition is that the factorial is the integer special case of the gamma function. The gamma function looks like this. As you can see, when you evaluate the integral, you'll get 0!=1. For the noncalculus savvy, check out the graph.
Its really ironic that they don't mention this in calculus classes because the students have the knowledge to actually work it out for themselves. Perhaps some do, but I didn't see the gamma function until college. Seeing it earlier might have really helped me with my plan to take over the world.
Every post in this thread, nay every post on Slashdot is redundant. I read them all in Pi yesterday.
Oddly, my ICQ number is at position 19724810 counting from the first digit after the decimal point.
What're the odds of *that*?
;-)
Cheers,
Jim in Tokyo
Have no clue about firewalls?
-- My Weblog.
My cat's breath smells like cat food
Bush should have died, not Reagan -- Morrissey
Morrissey rides a cockhorse -- The Warlock Pinchers
Is using the PI digit generators more random than using rand()?
> It will start a whole new branch of numerology dedicated to finding entire new holy books...
Ah, yes. Bible Codes for mathematicians.
I once seriously considered buying the Bible Codes program, just to see if it could find the message "bible codes, big lie". I wonder whether pi reveals its own naughty secrets.
At the very least we can expect helpful messages like "You can just round it off now, you moron."
--
Sheesh, evil *and* a jerk. -- Jade
> you should know one egregious example of funny strings in Pi at funny positions:
> 42424242 at position 242424.
Incredible! I have just discovered that it also lists out all the digits of pi, starting at offset zero!
Now instead of calculating the digits of pi, we can just look them up in the digits of pi!!!
On a serious note, observe that if pi does indeed have all possible strings embedded in it, then it must have all possible strings embedded in it twice. (And thrice, 4x, 5x, etc. The proof is left as an exercise to the reader.) Thus if it does embed all possible strings, it follows that the first n digits of pi must appear in it somewhere other than at offset zero, for any positive n.
The interesting question is, must it also have all infinite strings embedded in it? I suspect that would lead to a contradiction, but this goes beyond my mathematical competence.
--
Sheesh, evil *and* a jerk. -- Jade
Turing machines with attached "oracles" are used for proofs in theoretical computer science, but it's important to keep in mind that they are purely theoretical - no one will ever build an oracle. You don't think about how an "oracle" works. It's just a concept, like imagining that magic works. So why is this useful?
:-)
Well... For instance, you might be able to prove that such-and-such a problem with input size "n" could be solved in polynomial time (i.e. "fast") if you just had a magical oracle to supply you with only log(n)correct, one-bit answers.
The point of the proof would be that you don't need more than log(n) magic bits from the oracle. So what good is that? Well... If you can get the number of magic bits small enough, while still keeping the algorithm fast, it may provide a way to do a randomized algorithm where instead of using the magical oracle (which doesn't exist, remember?) you just use a random number generator, or maybe just try everything. Since a random number generator isn't as accurate as a magical oracle, you run the algorithm a lot of times with different random bits. Maybe you'll be lucky soon, and maybe that's good enough on average.
Theoretical computer science is fun. It's a little crazy and non-intuitive somtimes.
Torrey Hoffman (Azog)
Torrey Hoffman (Azog)
"HTML needs a rant tag" - Alan Cox
If there were an infinite number of monkeys typing on an infinite number of keyboards would they eventually produce all the works of Shakespeare? Not exactly - they would produce them immediately (as quickly as a monkey can type).
Not only that, but they would produce them an infinite number of times over and over.
Wow, I really should have previewed that one.
s/pit/pi
s/know the how/know how
and other grammer.
Wow, you just sparked a real idea, mathematitions say its impossible to encode a large truly random sequences of bits, into something that the outcome plus the decomressor is smaller than the first bytes. But given enough computing power you could find the large random bits in pit, somewhere and simple have the decompressor know the how to compute pi, and the start and stop points of the random number?
It is not hard to prove that essentially any number is a normal number - in the sense that any number you pick at random between say 0 and 1 (uniform distribution) (More precisely, the set of normal numbers in [0,1] is a set of full measure - one proof goes via the strong law of large numbers - ask your local probabilist for an explanation). What is hard is showing that a particular number is a normal number (I didn't even know that one had any explicit examples).
Normal numbers being essentially all numbers is more subtle than the fact that "essentially all numbers are transcendental". The set of non-normal numbers is actually uncountably infinite (not countably infinite like algebraic numbers).
Maybe pi is what happens if you take the place described in Jorge Luis Borges' "The Library of Babel" and run it through an encryption or compression scheme we haven't hit on yet.
The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries, with vast air shafts between, surrounded by very low railings...
jmac
Why concentrate on just pi? If they show it's true for all trancendental numbers, they've got it for pi, e, etc.
;)
I'd be happy with just pi for starters...
Furthermore, it is not true for all trancendental numbers: for example, 1/n^(1!)+1/n^(2!)+1/n^(3!)+1/n^(4!)+1/n^(5!)+...
are trancendental, but with n=10 that number has only 1's and 0's, so it's not normal.
Can pi appear in pi anywhere? I guess not, since that would mean that pi repeats. Could e be in pi? I suppose if e was in pi, and pi was in e, then pi would be in pi, which I guessed earlier it couldn't be. But, maybe I'm wrong and there's a loophole since if pi contains itself, there's an infinite recursion going on.
Pi can't appear in pi, because that would make it repeat and make it rational, which it isn't (at least the way I understand "pi in pi"). Is e in pi? Not neccessarily. These guys are trying to prove that any finite number series can be found in pi, and e is infinitely long. If it's true, then you can choose any n and you can find n digits of e in pi, but not infinitely many.
Of course, e might be in pi (though I consider that unlikely - this would mean that pi=p/q+e*10^(-n) where p, q and n are integers, which seems quite weird). But what these guys are trying to prove doesn't show that.
I doubt, therefore I may be.
If it has now been shown, then Nature (Ma Nature, not the journal) has given us the proverbial infinite monkeys, and I'm going to look for Shakespearean sonnets in that number. <g>
mefus
In Open Society, GPL Software frees YOU!
That pattern (1828...) breaks down after awhile, if that's what you mean.
mefus
In Open Society, GPL Software frees YOU!
I think that would make PI a repeating pattern...
so... IANAM, but I think that's been ruled out.
Damn, slashcode thinks I'm cowboyNeal, and won't let me post!
mefus
In Open Society, GPL Software frees YOU!
Well, you should list pounds vs kilograms, since nobody actually uses stones... (and to be more correct you should probably have pounds or stones vs newtons, since those are a measurement of force, and kilograms are a meaurement of mass...)
--
"It's tough to be bilingual when you get hit in the head."
The question would be if the messages are in ASCII or Unicode :P
--
"It's tough to be bilingual when you get hit in the head."
It was 60 seconds, then went to something like 90 though it still said:
Slashdot requires you to wait 1 minutes between each submission of comments.pl in order to allow everyone to have a fair chance to post.
It's been 1 minutes since your last submission!
I love the lack of granularity with that... the new 20 second Reply-->Post is interesting, too. Guess I need to learn to type more, or slower.
--
"It's tough to be bilingual when you get hit in the head."
The sarcasm (and my nobody tags) apparently got lost...
--
"It's tough to be bilingual when you get hit in the head."
Hi,
I'll try to summarize why this "pi compression" won't work, adn I'll try to take all of the mentioned improvements into account.
1. Finding the required string in pi
-> For every reasonable long string (say, the Max Payne ISO) this is gonna be a pain in the a**, obviously
2. Distributing the position
-> If pi is normal, the length of the starting position of the string would be as long as the string itself (on average), so no compression here
3. Compress the position with LZW (or your favourite compression algorithm)
-> Again, if pi is normal, the position of the string is likely to be truly random. Compressing truly random data is not possible. Even if the digits of the starting position are not TRULY random, the effect of LZW would be minimal if not eaten up by the algorithms overhead. There is no sense in distibuting a 649 MB number to get a 650 MB ISO...
4. Do the "pi-compression" until you get 1 digit (or reasonable few digits)
-> Won't work, because:
a) You "compress" string x and get string y,
which is the starting position of x in pi.
b) According to (2.) y is about as long as x
c) return to a), and remember (3.)
Sorry for the mess, but once again mathematics win over wishful thinking.
C. M. Burns
I suspect this would be slightly impractical. Assuming the Metallica MP3 takes up 3Mb... this is ~24 million bits.
The chances of any 24 million consecutive random bits being the same as the bits of this MP3 is 0.5 ^ 24,000,000.
This is a rather small number. For any digit of Pi, the chances of it being the start of a sequence that encodes the MP3 is a half times itself 24 million times.
I think any description of the computing effort required would involve phrases like "current age of the universe" and "quantum computers the size of the solar system". I suspect your distributed computing effort would be looking for a long time.
Why concentrate on just pi? If they show it's true for all trancendental numbers, they've got it for pi, e, etc.
Can pi appear in pi anywhere? I guess not, since that would mean that pi repeats. Could e be in pi? I suppose if e was in pi, and pi was in e, then pi would be in pi, which I guessed earlier it couldn't be. But, maybe I'm wrong and there's a loophole since if pi contains itself, there's an infinite recursion going on.
If there were an infinite number of monkeys typing on an infinite number of keyboards would they eventually produce all the works of Shakespeare? Not exactly - they would produce them immediately (as quickly as a monkey can type). That's infinity for you. The lazy 8. It goes on and on...
BigKahuna
Why ban it. Can't we just change it to someting simple like 3.000?
Comment removed based on user account deletion
So, does it contain the 7 line DeCSS implementation? how about just the ascii "Natalie Portman"? goatse.cx?
What am I trying to say? So what!
Well if it contains all conceivable messages, that must mean it contains all conceivable circumvention software! Down with Pi!
It's 10 PM. Do you know if you're un-American?
Not at all, people would just remember that the proof refers to "primes other than 1". The current definition was chosen so that you have to do as little writing as possible to describe what's going on.
The properties of numbers do not change, regardless of what names we give them.
The main reason for 1 not being considered prime is so that a theorem known as the "Fundamental Theorem of Algebra", is true when it talks about "prime" numbers. In natural numbers, the theorem amounts to the idea that each number has one (and only one) factorisation into prime numbers (eg. 140 = 2x2x5x7).
All your base are belong to pi
What was the topic -- Nerd Clue?
And now the the 0! question has been handled in the subthread, let's dwell on the paradox that is 0 to the 0th power.
We know that 0 to the anything-th is 0.
We know that anything to the 0th is 1.
So 0 to the 0th is....
Quoth the article:
The quest to conquer pi's infinite expanse has led some mathematicians into fierce calculating competitions. The current record, achieved with the help of supercomputers, is 500 billion digits.
500 billion digits? that's all? my nt box calculates that many instructions every time it boots. maybe the mathematicians should upgrade.
There is/was a German hacker's convention scheduled for this year, one of the topics for discussion that I haven't seen posted in /. is the 'illegal prime'. A prime number, when written in base 16, that becomes a .gz file with the deccs code imbedded within. Its about 1400 digits long, viewable here How long till a digit place and count to find it in pi becomes available? The smalled deccs code in c is about 430 charachters. Remove the CR/LF, encode as 7 bit, and it should be much easier to find inside pi.
With the predictability of pi digits outlines a couple of days ago, making a program to accept a place and length to output a planned file is very realistic. However, I believe we are far behind in the computing power to actually take an arbitrary file of more than a few bite's in size and find that location/length pair in pi. Let's hoep quantum computing changes that.
On an aside, all of the information on finding digits of pi are base 10. Are there any articles on predictability based on a binary representation?
Toodles
Toodles D. Clown
This is a bullshit proof. First, you define your number not to contain any 8s, and then you say "see, it doesn't contain any 8s!". But that doesn't tell us anything about wether or not there is a string of 5,646,498,765 8s in pi or any other irrational number in decimal.
EagerEyes.org: Visualization and Visual Communication
> and seven 7's is found at position 3346228, six 6's is found
> at position 252499, five 5's is found at 24466, and so on...
"and so on..." ?!?
That's not quite true. There exist numbers which have a PATTERN, but don't actually repeat.
.101001000100001000001...
For instance,
There's certainly a pattern (1 zero then 2 zeros then 3...) but the number never repeats.
Read: you're both pretty much right- it is a ratio of two numbers, and it is irrational (as odd as that sounds)
-bugg
so pi contains everything? information about everything?
so everything is out in the open. forget about privacy and the like.
Don't Panic
this is the latest : microsoft sues pi for containing the complete source code to windoze.
btw, the code starts at position 4200394298 (in the binary expansion of pi), and continues for well, as long as anyone ccan read the stuff...
Don't Panic
Doesn't matter how you do it, the computer converts all bases to base 2 (good old 1 and 0) which requires X amount of space. If you were to write it down, that might be a different story. Best (realistic) base would be about 58 (a-zA-Z0-9).
Rod Taylor
It's like saying that by throwin the dice more and more maybe you'll find a pattern.
Actually, I know the pattern there: I lose all my money.
---
python -c "x='python -c %sx=%s; print x%%(chr(34),repr(x),chr(34))%s'; print x%(chr(34),repr(x),chr(34))"
FLASH! All patents are declared null and void because all patents previously awarded have been found to exist within PI. Although mathematicians have not proven yet that PI has existed since the beginning of the universe, they have conceeded that not only has PI been around "a very long time" but that it has probably been around longer than Compuserve's GIF patents.
An anonymous scientist has even gone so far as to say it has probably been around longer than the human genome, rendering new drug company patents on human DNA void for the prior art contained within PI.
A roman catholic biship was overheard to have said, "Not only does this prove that there is a God, but it finally proves once and for all that he hates lawyers."
An anonymous lawyer who wondered if God's prior art could be nullified for His refusal to defend His prior art was killed by a stray lightning bolt in the middle of the Sahara desert in a highly unusual but unrelated incident.
All finite strings.
Say no to software patents.
Indeed. Better pick something like 42424242, which not only occurs way early (at position 242424), but for which not only the search string but also the position is an interesting pattern....
Probaility of it occurring so early should be less than 1% (we would expect it below 100000000, not below 1000000), and probability of the position being a permutation of the string is...well...amazingly small.
Small note for nitpickers: I counted the 3. as digits; the search engine does not. Hence the position shown is 242422 rather than 242424.
Say no to software patents.
Not necessarily so. If you define a (decimal) number as follows:
- it starts with 1.
after the dot, each digit at a prime number position is 1, and 0 otherwise
The number would be 1.01101010001010001...- The number would have no periodicity (because prime numbers become rarer and rarer), so it woule not be a rational number.
- It never ends.
- But still, by construction, it would not contain a single 8, much less a series of 5,646,498,765 eights.
Thus proving that never ending irrational number does not necessarily contain all strings. Btw, irrational numbers are always never ending, or else they would be a fraction whose denominator would be a large power of ten. Think about it.Say no to software patents.
If you did any math in your youth, you'd know that this is a perfectly valid way to do a proof. It's called "coming up with a counterexample". If somebody claimed that all prime numbers were odd, it would be perfectly valid to point out that 2 is both prime and even. Discarding the proof because "you purposefully picked 2 to show me wrong" is invalid, as this is the whole point of the proof.
Likewise, in this case DNS-and-BIND claimed that all infinitely long irrational numbers have necessary a long sequence of 8's in them. I refuted his claim by showing him a number which had no 8 at all inside. Now, what exactly is your problem with my refutation of that claim?
> But that doesn't tell us anything about wether or not there is a string of 5,646,498,765 8s in pi or any other irrational number in decimal.
You're right on that, but nobody claimed the contrary. Saying "not all irrational numbers have a long string of 8's in them" is not the same thing as "no irrational numbers have a long string of 8's in them".
It's like in real life: "not all pies are cream pies" (or expressed differently "it is a pie, so it has to be a cream pie"): indeed, there are also apple pies...
But that doesn't mean that "no pies are cream pies" (i.e. cream pies don't exist): indeed Bill Gates was hit by one in the face...
Say no to software patents.
42424242 at position 242424.
Oddly enough, according to the pi search page, the same string can be found again at position 1404114, which is also below 100000000. On a normal pi, you'd expect a single occurrance of 42424242 below 100000000, and at a completely random position...
Say no to software patents.
Cannot be, or else it would repeat itself an infinite number of times, cyclically, which would make it a rational number.
Say no to software patents.
Try the American version (mmddyyyy). God knows its harder and more complex than it needs to be, but I think it appears to be the standard that the universe works on....
Dijital
Diji
"I came, I saw, I WTF'd!"
Doesn't that make Pi illegal to distribute? I bet it can break e-book encryption too!
I can see the headlines now;
RIAA and DVD-CCA Join Forces, Battle Circles
Judge Patel Rules Depections of Circles Illegal
Government Passes Controversial "Digital Millenium Circle Act" Banning Distribution of Circles as Circumvention Devices
"It's now illegal, for example, to walk around a fence post designed to keep you out, even if there's no fence, since you'd have to walk a partial circle around the fence post, and the post, in the government's eyes, constitutes an effective measure to prevent access."
Actually it is possible. If you read an earlier post there is a string of eight 8s
The nature of Pi will not allow a repeating pattern of numbers, so long as no continuous pattern of those numbers appears to infinity, Pi can have every possible piece of information contained within.
Scary when you actually think about it
There's a signifigant difference between random and irrational. Random means you'll get a totall random value every time you look at it. Irrational means there's no pattern to it, while the number is most definately set and unchanging.
Hmmm....
128 was found at position 148
127 was found at position 148+149 (=297)
The second application should be the start digit of a Metallica MP3.
Anyone want to start up a distributed network to look for these?
I'm trying to teach myself to set people on fire with my mind... Is it hot in here?
I'm sure there's a lot of cool stuff in the first 2^128 digits. If 128 bits is a long on an itanium system, I'm sure we could have a lot of fun searching the first 2^128 digits of pi for stuff without even breaking out of the long address space. If 2^128 seems small, how about 2^256? 2^1024? 2^65536? That's not a lot of bytes, but it's a hell of a lot of space to search. Probably more than modern computers will be able to handle for years (Even if we do start up a distributed net type search engine to look for things.) Who knows. My next computer might have to include a pi coprocessor...
I'm trying to teach myself to set people on fire with my mind... Is it hot in here?
heh.. you can try, but it's common knowledge that those messages are really broadcast via TV static every night around 2:32am.
So, yes, trivially, all strings appear an infinite number of times. Or are we talking about another measure of frequency (number of appearances in a substring of pi's digits of a given length?)
There's 10 types of people in this world, those who understand binary and those who don't.
Eight 8's happening early enough in pi that we'd notice is extremely unlikely, as we can all imagine. This makes it, on first inspection, pretty damn cool that it happens. But then, when you think about it a little more, you realize that while eight 8's is unlikely, "something that humans find interesting" is very likely, mostly because we find so many strings of digits interesting. From that point, it's just random which particular interesting string crops up, since we know one is going to, eventually.
This is a self-referential sig
One of the weirdest facts about pi that i've ever heard is the following: the length of a sailboat, in feet, divided by its hull speed (the maximum speed a boat can go, at which point its bow and stern waves cross so that it can no longer accelerate without planing), in knots, is, you guessed it pretty damn close to pi! Now, by "pretty damn close", I don't mean by an irrational number researcher's standards...it's more like 2 or 3 decimal places...but to a sailor, that's close enough that the "pi" button on the galley's calculator works perfectly.
This is a self-referential sig
It will start a whole new branch of numerology dedicated to finding entire new holy books... the Book of the Damned, I Microsoft, II Microsoft, the letter of BOFH to the Great Unwashed, and, of course, the source code to Office (which will take up the space between 2^8 and 2^40906) ...
Zaphod B
Zaphod B
When duplication is outlawed, only outlaws will have
So if anything, they are proving normality to base 2^n, NOT base 10^n. And it may actually be that their proof is general enough to show normality in all bases - the article is not clear on that point.
If every known string would be found. They what about finding Pi with one digit off? Like Where we found Pi and every digit in pi is the same except say the trillionth digit. Now. Start your recursive looop engines and figure out the rest
--------========+++Dont Feed The Lab Techs+++========--------
Unfortunately, you need to specify the index to the beginning of the message. Since your message is probably a very long way into Pi, the index will likely be more bits than the message itself. I discovered this property of compression when I tried to build a compression routine based on Godel numbers. Most compression algorithms use the assumption that there are repeatable elements that can be compressed. This is sort of like using fewer bits to represent the low frequency component of a signal. If the message doesn't have much repetition, that algorithm will do poorly. You could also write a compression algorithm that knew commonly used phrases - this algorithm will only work well on phrases in the domain for which it was designed. All of this is related to Wolpert's No Free Lunch theorem.
You're right that we're likely to find a few interesting sequences earlier than we expect. However, this isn't one of them. Finding 88888888 (~8.9*10^7) at or before position 46663520 (~4.7*10^7) is clearly not unlikely. It should be around 37% probability.
I don't want to go off on a tangent about proofs... but I'm curious - what happens when a rule of mathematics is challenged? For example, some defenitions seem so arbitrary to me. 1 is not considered a prime number because it has only itself as a factor... I don't like that reason and I don't kow why. Years of work would be invalidated should such rules be thrown out, right?
Uh, changing a definition is hardly throwing out a rule. If you don't like one being prime, just take any mathematical work and replace every reference to 'primes' with 'primes other than one'. Whether you like how things are named has nothing whatsoever to do with validity. Definitions aren't really arbitrary, but they could be without messing anything up.
Ok, if an average MP3 is 5MB (5,242,880 bytes) then the odds of finding a specific MP3 using a sequence of random numbers is 1/256**5,242,880 (since 8-bit bytes have 256 possibilities). This is about the same as 1/10**12,626,113 (since our base 10 numbering system gives each digit only 10 possibilities)
P(MP3)=1/10**12,626,113 (really close to 0)
Thus the odds of not finding a specific 5Mb MP3 using a sequence of random digits is 1-P(MP3)
~P(MP3)=1-1/10**12,626,113 (much closer to 1)
Since the longest known expansion of Pi is about 500 Billion digits (500,000,000,000) there are 499,987,373,888 consecutive strings of 12,626,113 digits contained in the known expansion. So the question is, what is the probabilty that at least one of these is the specific 5MB MP3 we are looking for.
An easier question is to ask what is the probabilty that we won't find the specific 5MB MP3 in a 500 Billion digit expansion of Pi.
The probability that any specific 12,626,113 digit substring of the 500 Billion digit expansion of Pi is not the 5MB MP3 we are looking for is ~P(MP3). So the probability that every one of the 499,987,373,888 possible 12,626,113 digit expansions is not the 5MB MP3 we are looking for is ~P(MP3)**499,987,373,888.
P(~MP3)=~P(MP3)**499,987,373,888
So now that we know the probability that a specific 5MB MP3 file is not contained in the 500 Billion digit known expansion of Pi, we can calculate the probability that we can find at least one instance of a specific 5MB MP3 as ~P(~MP3)=1-P(~MP3).
~P(~MP3)=1-P(~MP3) =1-~P(MP3)**499,987,373,888 =1-(1-P(MP3))**499,987,373,888 =1-(1-1/10**12,626,113)**499,987,373,888
Hmmm... I think I'll go by a lottery ticket.
Mechanik
Last time I checked it started with
3.141
As always.
Of course, the real problem was that the required precision required the location be controlled to much less than the width of a carbon atom (and probably was beyond the reach of even the fundamental smallest distance possible in this universe - yes, there is such a distance).
That's why they call it science fiction - can you imagine a typical slacker hollywood type ever understanding such a fine point? Of course, hollywood types can't understand how geeks just do not seem to "get it" that understanding subtle film angles can make a movie infinitely more enjoyable either. To each their own, peace.
If pi were repeating, it would be rational (expressable as the ratio of two integers). The proof of irrationality of pi is somewhat complicated however. If you want to see an easier example of how one can prove a number to be irrational, consider the irrationality of the square root of two.
Other possible things that we could find -
PI IS THREE!!
froin laven, i didn't think i'd have to use that.
sig?
Then there are the 'dirty' irrational numbers like pi and e that seem to have random digits. The research mentioned has moved a big step closer to proving that the digits of pi don't just seem random, they truly are random (at least in the sense that all possible combinations occur).
The part that'll really blow your mind is that somebody found an equation that tells you any binary digit of pi you want, without having to calculate any of the other binary digits. (See here.) That is why people are excited by the conjectured normality of pi: if normal, it produces all possible strings of bits from a trivial deterministic equation. This mixture of randomness with order is at the heart of many interesting questions in chaos theory, computational theory, and cryptography.
-- ;-)
Kuro5hin.org: where the good times never end.
And that is a pain in the neck for everyone in comp.compression.
There is a frequent fallacy among those who almost understand how compression works, that works like this:
The assumption, of course, is that the number of digits and the offset can be encoded in a form that will be smaller than the original sequence. There is nothing to warrant that assumption. The fact is that the number of possible inputs that a lossless compression method can handle places lower bounds on the average length of its outputs. This means that no lossless compression method can achieve a lower average length for its outputs than would be achieved by simply numbering them all with the non-negative integers.
In fact, 'compressing' a sequence of digits into a (length, offset) pair will do substantially worse, since there are multiple (length, offset) pairs that will correspond to a given digit sequence; for instance, "1" could be encoded as (1,0) or (1,2). This duplication means that (1,2) is essentially wasted, since it could be representing a sequence that currently has a longer representation.
Lossless compression methods need to be used in conjunction with models: some criteria that separates the data we will want to compress from the vast majority of files, about which we do not care. The accuracy of this model affects how many of our inputs we can actually compress, and its precision affects the average compression ratio.
If people are to respect the law, perhaps the law should begin by respecting the people.
Of course, if this article is right, then we'd expect to find that picture somewhere, because it'd be just as likely to occur as any other.
Of course, he's technically wrong -- it could be a statistical fluke.
But the real point is that the god of that particular universe (the author, Sagan) put a message that probably isn't found at that point in pi. So all Sagan did is prove that his book was created by a sentient being.
This is the exact same story as last friday
http://slashdot.org/article.pl?sid=01/07/27/133823 %200
Euler was a cool guy... among this formula he was also the first iirc to prove that:
sum( 1/n^2, 1, inf ) = pi^2/6
of course, now adays that isn't too hard, as long as you accept that fourier series are valid, you can choose the propper function, expand it, and then evaluate it at a certian point to get the same result. But Euler didn't do this... he did an infinite factoring of polynomials! I can't remember which one at the moment (dohh...) my notes are at home, and I am at work
Of course, pi isn't my favorite number, gamma (euler's constant) is. It is defined as...
gamma = lim( sum( 1/k, k, 1, n ) - ln( n ), n, inf )
if you spend a little time you can prove to yourself that this limit does indeed converge. The cool thing about it is that there arn't any algo's that can calculate its digits, so proving that it is simply irrational is a hard task (i don't think anyone has done this yet). Although I must say, it isn't as useful as pi, but I still think it is coolerI post to slashdot in order to have your advice.
Everyone here wants to find the book of the dead or the decss source code or the max payne iso by getting a formula for pi digis and the offset, but don't you also need the length of the content? Or is that not necessary?
Peace,
Amit
ICQ 77863057
[o]_O
What if PI is a rational number, with infinitely large numerator and denominator? :)
The REAL sam_at_caveman_dot_org is user ID 13833.
Google couldn't find it, but it could be found with another word with a double o
God is real unless declared integer
Pi is not that unusual. Here is a simple number called (I believe) the Champernowe Constant
0.123456789101112131415161718192021222324...
After the decimal point we are, in effect, counting. Clearly, any string "35002134" will appear in the Champernowe constant, and infinitely often. (Anybody know where I got the sample string?)
Pi is an amazing number, clearly, but sometimes it is erroneously represented as the only number with the above property.
DJS
http://www.dougshaw.com
God is real unless declared integer
"Mathematicians have known for more than two centuries that the number is an infinite, non-repeating decimal. "
Why limit yourself to decimal? Base 10 was your grandmother's number system. Why not evolve and use binary or hexadecimal?
It seems to me that normality in one base would indicate normality in any other base. Anyone care to prove/disprove this?
You can never equivocate too much.
Ahem, does nobody remember that we had this very discussion just a couple of days ago? Some interesting points were made there, perhaps you should all have a review before spitting out the same lame jokes in this thread as well (those relating to copyright and DeCSS and privacy, for example).
"If you think education is expensive, try ignorance" - Derek Bok
Don't you see that since every string appears somewhere in pi, this would make a great standardized naming scheme for every string? So the current US population (284,804,918) occurs at position 68768290872 in pi. The point is, all possible strings are in pi, and in many other irrational numbers, apparently.
Super Encryption:
Your key would be some irrational number, and the encrypted message would be a position value in that number (plus a value about how many digits are relevant after that position). Decryption works like this: you write out the irrational number which is the key, go to the position indicated, and start reading the message! Simple! But literally impossible without the key, because then you wouldn't know what irrational number to look at (and there are aleph-one many); it's not like this would ever be crackable; no quantity of CPU power would help. (There are practical concerns; it's pretty computer-intensive to write out binary-expansions of irrational numbers, so if the messages were long, you'd have to go pretty far out to find the position, which is impractical. But this might be a neat and practical way to encrypt the decoding key for a file!)
Quick, patent this before Rambus people read this far!
When somebody asks for my phone number in a bar, I'll say: you'll find it at digit 20684081 in pi. If they call, I'll know it's serious!
Man, the whole 'find any message you want in pi' is like, such old news. I pointed out the relation between randomness of its digits and message encoding a whole week ago!
Of course, this just proves once again that I always get around to replying to stories too late for anyone to notice.
-----
Free yourself. Everything else will follow.
The middle mind speaks!
Do you have a constructive proof of this oracles existence? Constructive is the key word here. The problem is similar to finding the (Kolmogorov) complexity of a natural number. You can prove that every number does have a complexity (measured as the minimum size representation of that number), but you can't compute the complexities. The function exists, but you can't compute it.
Turing's theorem is much more interesting that you seem to give it credit for. Let B be a set of natural numbers. Define the "B-Halting Problem" as Given an arbitrary Turing machine, determine whether that Turing machine, when equipped with a B-oracle, halts after being started with empty input. Then Turing's proof shows that no Turing machine equipped with only a B-oracle can solve the B-Halting Problem. The usual, oracle-less, version is a special case of this: just take B = empty set. You're absolutely right. The Turing Theorem is interesting (as long as you don't consider the original paper. . .snore!). It's unfortunate that everyone just chooses to ignore it, instead of really investigating what the implications of the theorem are (along side with Godels Incompleteness Theorem). They're mostly held up as clever examples, then discretely swept under the table.
Randomness is not an essential part of the study of Turing machines-with-oracle, and the Halting Problem does not involve randomness at all.
Not at all? The fields share many similarities, and it's hard to find a modern discussion of one without the other.
Cheers!
The middle mind speaks!
I saw an excellent cartoon strip where a guy lves in a world where everything is square. He buys a package from a shadey guy, tucks it under his overcoat and smuggles it home. When he gets home, he opens it, takes out a pair of compasses and starts to draw circles. If anyone has a URL to it, id love to see it again...
Offtopic, Inflammatory, Inappropriate, Illegal, or Offensive comments might be moderated up.
A teacher, a physicist and a mathemetician are having drinks together in a Scottish pub when the teacher looks out the window and sees a white sheep. The teacher says, "There are white sheep in Scotland". The physicist looks out the window and declares, "There are sheep in Scotland; we have already detected and confirmed white ones." The mathematician says, "In Scotland there is at least one sheep, at least one side of which is hite."
No, I didn't pull that from Singh (it's there, though). It's an old mathematician's joke but it's true. The most anal Rainman you've ever seen is incredibly chaotic compared to mathematicians (at least when they're working on a proof or theorem).
woof.
"No ma. You don't have to worry about Code Red. Yes, I know CNN told you that you do. Ma, do you run a Web server? No, Netscape is a browser, not a server. Yes, there's a difference. You don't want to know. No, and the Internet didn't die last week, either..." -- my side of a phone call two nights ago.
Interesting link for searching pi. I tried every 7-digit phone number in my head (my hom enumbers, work numbers, friends, etc), and it found all of them! Spooky.
BroadbandPig
Consider one of the most basic uses of the factorial function - determining the number of permutations of n distinct objects. n distinct objects can be arranged in a line in n! ways. This is easy to see if you have 2 or more objects. If you have one, then you have no choice in how to arrange it, since 1! = 1. If you have no objects, you still have no choice in how to arrange it, thus 0! = 0. That's one explanation of why it's defined that way.
ACK! That's supposed to say 0! = 1. (Next time I'm I'll actually read the preview.)
> Sure, if you have 1 object you can arrange it in only one way. But if you have no objects (zero), there is nothing to arrange. Nothing to be done.
Yes, there's nothing to be done, and there is exactly one way that you can do nothing. How many ways can you arrange zero books on a bookshelf? There's no choice there, and hence only one possiblilty - your shelf is empty.
And the Scotsman says, "You lads keep yer filthy, stinkin' eyes off me wife!!!!"
(Sorry, couldn't resist...)
Of course, finding the appropriate sequence would be the challenge...
Pi is an integer in base pi...
"...not a single message but every conceivable message, meaningful or not." ...like my voicemail?
Katsuyo Mori
Does this mean if you sat down a bunch of monkeys in front of a computer, and had them calculate pi forever, you'd wind up with all the works of William Shakespeare? ;)
I don't know where you saw it, but it was a troll :)
Let's say you want to compress 1024 bits messages, which is of little use, but let's say just 1024.
As you would want to "compress" any 1024 bits messages, unless you are extremely lucky about overlaping, you can't expect them to be very close each one of the other, in fact, the larger your message, the lower you'll have a probability to find it "soon"
But even then, let's say you are so lucky, possible message 1 does starts at digit 1, and possible message 2 does start indeed at digit 2
Well, you see what I mean, just to tell someone you are sending him message number xyz, you will need a 1024 bits adress.
Of course, you didn't need PI to do that, you just take the message as a number, and you send you message itself as being a number and you message is message number "the message"
Which is of course no compression at all
Using PI, you have the garantee that it will be much worse than that as the message won't be following each other as you'd like them, so that a 1024 bits message "xyz" might be found at an address so far away you actually needed 4096 bits to give its position.
how many bytes did you need for "pou" and how many bytes did you need to tell us you can find it at address 4602166 ? the same number...
And we didn't talk about also saying "and it ends after 3 bytes" which you must code also, unless you send a "pou\0" but it will probably be much worse then.
It's even less useful than that
if you just have to recompute PI, well so far the compression part is so far atractive.
But just think about how huge is the "position"
That is, you ound your 4 characters string in PI, ok, but where ? you will give the position instead of the string itself, but remember the position will be a number so huge (sometimes, of course, not every time, but for any practical application...) you will need more than 4 bytes to represent it... so far for the compression..
Maybe all pies are just implanted into our heads!
So there, cream pies may exist, but they may not exist also.
If you would like to prove me wrong could you please bake me a bunch of pies and then mail them to me? Along with money? Ok ok, just the pies then.
-Gnight
I agree, ever since taking Calculus I have question why is 0! (zero factorial) equal to one?? I asked a professor in my universtiy once and his reply was, "It just is, if you dont believe it, solve the problem without using it". Sure, I used 0!=1 for the problem, but I still doubt its validity. Maybe they made it up so theorems like "Taylor Series" make more sense.
Everything is a conspiracy.....Sure, if you have 1 object you can arrange it in only one way. But if you have no objects (zero), there is nothing to arrange. Nothing to be done.
Another, proof they use to prove this is as follows: n!= n(n-1)! 1!=1(0)! Therefore, 0! = 1.
Sorry I cant buy that. They say the definition of a factorial of a number n is the product of all of the positive integers from one, up to n.So my argument is this: if n=0, since we are trying to find 0!. How can you find the the product of all positive integers from one up to 0. This would lead you off into infinity.
My conclusion is that they made 0!=1 bcos it would screw the hell out of all the theories they already have.
I told you it was all a conspiracy
so basically we are using 0!=1 for convenience...that's great! so all the times when i couldnt find the answer i could have just made up a special purpose definition or theorem to make my analysis "work".
Ok. The standard solution for solving e^(i*A*X) was useful for solving second-order differential equations (ordinary form: P(x)y" + Q(x)y' + R(x)y = G(x) ) . My notes are sparse and my brain is tired, so I'd rather not try to remember the whole series of steps required to get to the part where this is actually useful. If you want the long version of the proof, ok, I'll post it tomorrow if you say so.
I'm not even sure if this was derived in class or just one of those few equations that were given to us "just trust me" sorta things. I had the bad habit of rarely writing down proofs, so I'd probably have to hunt this one down online or in my of my Calc books.
But, magically, here in my notes it says:
e^(i x B[beta] * x) = cos (Bx) + isin Bx .
My apologies for not desiring to hunt down the appropriate symbols.
So, with B = Pi and x = 1, you get:
e^(i*Pi) = cos Pi + isin Pi .
The cosine of Pi is -1 and the sine of Pi is 0, so it becomes = -1 + (i)*0 = -1 . Notice that e^(-i*Pi) also gives -1.
So in summary it really wasn't much of a feat for me to reproduce the final parts of the proof, only a matter of remembering what that standard equation was (shortcuts are wonderful things). Let me know if you really want me to derive the top part though.
"The universe seems neither benign nor hostile, merely indifferent." --Carl Sagan
Thanks. I always knew that not writing down the derivations in class would catch me at some point =).
Well, in one case they already did. In my Calc III class my professor proved a relatively simple theorem and promptly put that theorem on the first page of the test. I wrote the theorem down, but I never really committed it to memory. That was one of the few tests I got a B on...oh well.
"The universe seems neither benign nor hostile, merely indifferent." --Carl Sagan
Trying to imagine why every n digit number shows up the exact same amount of times is hard to imagine at first. But then, once you think about it, on an infinite scale, it would seem to attest to Pi's true randomness.
On a side note, I had a Calc II professor awhile back that wrote on the board:
e^(i*Pi) = -1 (of course, using the real symbols).
Then, he proved it. I have the proof written down in a notebook and I even managed to work through the final parts of the proof (it uses a standard solution for finding e^(i*A*X) without using it. If anyone is really interested in seeing it, I can post it (in rough ascii math =) For those of you with TI-92s that don't believe me, type it in. That magical machine can do more than I give it credit for sometimes.
Anyway, I just thought it was absolutely incredible that you could mix the two most popular transcendental numbers with the imaginary number (square root of -1) and spit out plain old -1.
"The universe seems neither benign nor hostile, merely indifferent." --Carl Sagan
but anything infinite already contains everything finite.
It's just the signal to noise ratio get completely ludicrous.
'There is a Light that never goes out.'
If infinity contains all finite things then surely it has to contain them in equal proportion?
Granted this approach isn't as useful as theirs for other things.
'There is a Light that never goes out.'
Well, then I gues this story is a duplicate post
A (rough) definition goes as follows.
Suppose B is a subset of the natural numbers N = {0, 1, 2, 3, ...}. Then a B-oracle is a device
which can correctly answer any question of the form
"is x an element of B?", where x is a natural
number. By "answer" we mean that given a
natural number x, the oracle will in finite
time answer "yes" or "no", depending on whether
or not x is actually in B.
For some sets (such as the set of primes) we can build an oracle easily: just write a computer program to do it. For the primes the program could look something like this. The number, x, is given:
- If x = 1, answer "NO" and stop.
- Set d = 2.
- If d * d > x, answer "YES" and stop.
- If d divides x, answer "NO" and stop.
- Increment d by one.
- Goto step 3
We don't care that this algorithm is horrifically inefficient: we are just interested in the fact that there is an algorithm to recognize the primes.For most sets though, we can't write a progrem to do this: under the generally accepted notion of "computability" (which comes from Church's Thesis) only countably many subsets of N are computable in this way, while N has uncountably many subsets. In these uncomputable cases, an oracle is assumed to exist as if "by magic."
Now, suppose M is a Turing machine equipped with a B-oracle. This machine is just like an ordinary Turing machine, except it can ask its oracle any question of the form "is x an element of B?".
[One way to imagine M is to assume it has a second, write-only infinte tape, on which is written an infinite string of zeros and ones. The n-th digit is one exactly if n is an element of B. Then, to "consult the oracle", M just reads the appropriate digit on its second tape. Any specific digit on this tape can be reached in a finite number of steps ("finite time"), so the second tape, along with some read logic in the machine itself, acts as the oracle.]
Further suppose that M "recognizes" the set A, which is itself a subset of the naturals. That is, suppose that M acts as an A-oracle, able to correctly answer any question of the form "is x in A?". In this case we say:
This relationship is written A <=_T B (where the underscore represents a subscript, and the ugly '<=' is meant to represent the "less than or equal" sign.)If A <=_T B and B <=_T A then we write A =_T B and say that A and B are Turing equivalent. From the stand point of computation, A and B are essentially the same.
However, the real interest lies in the relation <=_T itself. This relation turns out to give an extremely rich structure to the subsets of the natural numbers, and is the subject of much mathematical research. The modern standard reference for this field is Soare. This is targetted at graduate students and upper-level undergraduates. Other good references are Rogers (warning: Amazon link) and Davis. Davis is rather out of date now, but it is published by Dover and is cheap.
It might seem that talking only about the natural numbers means that none of this is very interesting. However, with the use of Goedel numbering, any finite sentence over a finite alphabet can be encoded as a natural number, so Recursion theory applies to all sorts of structures. One example is the set, X, of (oracle-less) Turing machines. Turing showed that the subset of X which consists of those machines which eventually halt after being started with an empty tape, is not computable. This was his resolution of the Halting Problem.
Cheers,
quokka
Indeed, let K be the set of all (Goedel numbers) of (oracle-less) Turing machines which halt when started with empty input. Then a Turing machine with a K-oracle can trivially solve the Halting Problem: it just examines its input (which will be the Goedel number of a Turing machine) and checks if it lies in K. If so, it answers "yes" and otherwise answers "no".
Turing's theorem is much more interesting that you seem to give it credit for. Let B be a set of natural numbers. Define the "B-Halting Problem" as
Then Turing's proof shows that no Turing machine equipped with only a B-oracle can solve the B-Halting Problem. The usual, oracle-less, version is a special case of this: just take B = empty set.Randomness is not an essential part of the study of Turing machines-with-oracle, and the Halting Problem does not involve randomness at all.
Cheers,
quokka
Indeed, this is the whole point of using an oracle: it allows the Turing machine to access information that it could not possibly compute for itself. If we restricted oracles to effectivley computable sets then we may as well not use them at all: just replace the oracle with a subroutine that computes the oracle set itself.
It is a mistake to conflate Computability Theory (also called Recursion Theory) and Complexity Theory. The latter is the more important to CS, and involves the analysis of algoriths for efficiency. The former is more abstract (as the use of uncomputable oracles suggests) and concerns itself with questions of the existence (or otherwise) of algorithms, but not their efficiency.
In general, this sentence is not true. I have had very little exposure to the theory of randomness, so I can't make any useful claims about the similarity of it to Computability theory.However, the Halting Problem, as a pioneering result in Computability theory, is quite independent of randomness. Computability Theory is not the same as Complexity Theory (in which statistics plays an essential roles), and it has a large literature which doesn't mention randomness at all.
Cheers,
quokka
...for every copyright protection scheme ever invented! And for every one that ever will be invented, for that matter. Not only that, it contains my entire illegal mp3 collection (and yours too).
Am I in violation of the DMCA every time I divide the circumference of a circle by twice its radius? Hmmmmm....
I wonder if this is what the menaning of like went to after the sales reps crashed on earth ????
--
The computer told me to press any key to continue,I pressed the one looking like this (|) !!OH SH*T!!
I always thaught that "1337" looked wrong "31337" look better to me
--
The computer told me to press any key to continue,I pressed the one looking like this (|) !!OH SH*T!!
I'll assume that this post was a joke, because if it was real, it's the most ridiculous advice I've ever read. Are you going to suggest that he go to BeOS the first time his Netscape browser crashes in Linux? How about you offer a suggestion on how to fix his browser in his current situation, instead of putting him through an OS Install to fix a browser crash.
{sigh}
DrPascal: Not the language, the mathematician.
Before his untimely death, Alan Turing did a lot of work on a new theoretical machine capable of transcending the limits of conventional Turing machines. The new machine (the so-called O-machine), would be a Turing machine connected to an "oracle", which would store some irrational quantity that would be able to do things like solve the Halting Problem, since it would contain an infinite amount of information (including overy possible program that could be created). At least, that's as much as I can remember (no links, sorry).
Who knows, maybe pi would suffice for such an oracle?
Of course 0!=1
...
just try it out in gcc!
if (0!=1)
printf("hehehe");
Those whose signatures threaten negative moderation will be modded down.
that pi is totally and completely random?
turn up the jukebox and tell me a lie
Sounds hairy. *cough*
--
The internet proved this one wrong.
Woudn't truly random mean we can't predict what the next number will be? By the very existence of the formula to calculate pie digits, Pie can no longer be random. Maybe if falls into that pseudo random category, where all the random number generators come from.
I remember when I was working on engineering design that PI=3.1416.
That was good enough for us.
Silly.
A string of 1 1's is located at position 1.
So according to Contact, embedded into the digits of Pi is the picture of a perfect symbol.
Err I mean "the picture of a perfect circle."
"And like that
Not being a troll, but I still don't see the big deal about one irrational number.
In Carl Sagan's book, Contact, there is an interesting revelation made to Ellie by the alien she visits light years away. It tells her that buried deep in Pi is an important message.
(Here's where my memory gets a little iffy.)
So when she returns home, she writes a program that searches for non-random data in Pi, in multiple bases, and sure enough she finds a message in base-11 composed of all 1's and 0's.
When laid out in rows of equal columns, a perfect circle is formed out of 1's, with 0's as the background.
So according to Contact, embedded into the digits of Pi is the picture of a perfect symbol. If this were true, it would be proof that the universe was created by intelligent life.
Or at least a real funny joke.
"And like that
To be complete:
...
9 position isn't in 9 digits
8 does
7 does
6 does
5 does
4 doesn't (5 digits)
3 doesn't (4 digits)
2 doesn't (3 digits)
1 does
I think it has more to do with the probability of a n digit expression to appear
My birth date (1711981) doesn't appear in the first 100,000,000 digits in pi. I think I don't belong to this universe...
The digits 12345 occur at position 49702.
That's amazing, it's the same as the combination on my luggage!!!
(Spaceballs in case you didn't catch it)
Well, as others have rightly noted, this solution wouldn't work. It takes N digits to represent a number of N digits, quantum mathematics aside, as long as those digits are more or less random. Compression programs like ZIP only work at all because certain strings of numbers are more common than others in computer files (if I understand the technology correctly).
However, this idea could go the way of all complex (yet failed) compression algorythms: encryption! Imagine trying to decode the resulting index, with no idea that Pi was even involved. Not gonna happen.
I can't say the idea didn't intrigue me for a few seconds, though; adding infinity to any equation always makes for the most fascinating possibilities.
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This article reminds me of a discussion I had w/ one of my friend about burning a huge number of cds with every possible combination of bits that can be contained on a 74min length cd. The result is really mind boggling. You can have say a cd containing, pictures files of your future kids, wife, anyone imagineable, in jpg, bmp, or any other file format. Same can be said with video, audio, movies, games, software. Imagine the next hit album of , the next matrix and star wars movie (in vcd format only tho =) did i also mention doom3 and quake4, or even porn of ! Its really weird when you think about something like heh.
i think i should get some more sleep heh.
"All finite strings. "
Does this mean that sooner or later, someone will be able to prove that AYBABTU can be found in the binary digits of PI?
And will it give them more or less karma?
And both the MPAA and RIAA would go under because CDs, DVDs and film reels would be outlawed. (Because they're circular.)
And the IOC would be gone too...think about the Olympic Rings!
We'd even be able to burn all those ancient BeeGees disco LPs...
Maybe this won't be so bad after all...
I remember watching Northern Exposure when I was about 13 and there was this episode where Chris Stevens dates this mathematician chic and she talks about a string of eight 8's. Years later when I read about a Pi search engine I tried it and was actually surprised to see it worked.
/. = useless posts.
alcohol +
:o)
BOSTON SUCKS!
For instance, the number 15485741351654 in decimal converts to e158ecc2e6 in hex (base 16). Now, this is a 4 character saving in plain text. If you used unicode and a large base (say, to the order of 2^9 or something) to encode the number would be much smaller, and thus would only have to be converted to decimal, located in pi and the message read. The large base could be a standard (because arbitrarily selecting and transmitting a very large base could be as cumbersome as putting the number in decimal in the first place).
Overcaffeinated. Angry geeks.
We take the word pou and get it's binary form (in ASCII). This is 011100000110111101110101 then we convert that to decimal, getting 7368565 and then we search pi for that number, which we do find (yay!) at digit 4602166
Now all we have to do is transmit the digit number and the number of digits after it you have to read. That would be something like 4602166/7 Granted, that's not much of a compression system, since the whole message takes up 9 bytes in ASCII and the "decompressed" one (pou) takes up only 3. But, given larger messages, and converting these numbers to a large base (how many characters can unicode represent?) this could be a really useful compression system, provided you can indeed find the sequence you are looking for in pi (which is really hard right now)
Overcaffeinated. Angry geeks.
Not very nice to reply to my own comment, but I just thought of this: When using a very large base, you have a symbol to represent each character. Granted, this is stored in n bytes, which increase as the base does, thus making this impractical, but what if you were to make a .gif out of it? you could generate the very large base number, make a gif (or any kind of image for that matter, preferrably 2 bits per pixel) of it, and send it over. There's a base number (would have to calculate that) where this would actually make sense (the image representation of the character takes up less space than the computer representation of it). Just a thought.
Overcaffeinated. Angry geeks.
0.31331333133331333331333333133333331333333331... (where the ones get further apart each time) is irrational, but it isn't normal, and the digit positions are trivially predictable. You're confusing normal numbers with irrational numbers and thinking they're the same thing. Not every irrational number is normal.
Numbers in large bases require more bits for storing each digit. And if you're wanting to involve a computer in this, any base representation you pick will ultimately be converted to binary anyway.
Or take an other example: There are infinite many natural numbers (1,2,3, etc., sometimes starting with zero, depends on which is more convenient). But not every number you can imagine is natural.
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use Bielefeld.pm
Doesn't it mean that somewhere in there the value of pi will be stored, meaning it suddenly has become a repeating string of digits and thus not random anymore?
just wondering...
bash$
does this not imply that while pi may contain the complete source code to Office 2000, it also contains all possible incorrect versions, and it is impossible to know which one you have found.
:-)
impossible? not really, we already have a bunch of them
bash$
Every mathematical "definition" is arbitrary. The logic in mathematics lies in the derivations from the definitions, not in the definition themselves. You may have useless definitions (which are in contrast with "usual" maths), fruitful definitions (which let people find a lot of new things), or boring definitions, which do not give a lot of interesting results.
In the case of factorization, 1 is called "a unity", that is something separated from the prime numbers, because the most important result is the unique factorization ( up to unities!!!) in the group of integer numbers with the usual "multiplication" operation.
Notice that there are also a lot of theorems which have to distinguish the case of 2 and of another prime: the reason is not that 2 is the only even prime number, but rather than in the field generated from the first 2 numbers, "1" and "-1" are the same thing. But those results are not so important to deserve a modification of the definition of prime numbers to cast out 2.
A last comment: it is not difficult to write down a number which is normal in base 10. It suffices to digit 0.1234567891011121314151617181920... But if someone ever shows that \pi (or e, or a "common" irrational number) is normal in a certain base, it would be a really important result - for mathematicians, that is. The practical importance would be zero.
ciao,If a five-hundred digit sequence of numbers can be represented instead by, say, a twenty-five digit offset value, that seems like quite good compression -- until you remember that the only way it would work is if either: the (de)compressor carries around a datafile of "all" (well, a huge chunk of) for reference, or it has to recalculate up to the position of that offset on each run. And there's where it all breaks down.
Liberty in your lifetime
0123456789 : from 17,387,594,880-th of pi
0123456789 : from 26,852,899,245-th of pi
0123456789 : from 30,243,957,439-th of pi
0123456789 : from 34,549,153,953-th of pi
0123456789 : from 41,952,536,161-th of pi
0123456789 : from 43,289,964,000-th of pi
Liberty in your lifetime
Liberty in your lifetime
Some have suggested repeatedly applying this algorithm, to generate successively shorter indices, which can "compressed" in turn by the same method, but this will not work for several reasons. Here's just one:
Let's accept the assumption that the index I of an N-bit message will typically be about N/2. N/2 is only a single binary digit shorter than N. Not much of a compression scheme at the best of times!
[To be exact:
Index(50%): k = ln(2)/ln(n/n-1) from the Poisson Eq.: [(n-1)/n]^k = 50%
This factors in the odds of hitting undesired repeats before the desired match.]
However, it is not enough to specify the index, you must also specify how many digits to read beginning at that index. So, instead of an N bit message generating (on average) an N-1 bit index as output, it would generate a pair of numbers (Index, Length), with the Index typically being N-1 bits long, and the Length typically being log(base 2, N) bits long.
Therefore, the output is typically N+log N-1 bits, which is greater than N for all N>2, The "pi index" method does not usually compress the data at all! Quite the opposite!
Are you sure you know what you're looking for/at? Is the number 42424242 one that you would search for first, or a pattern one found after the fact? These are VERY different things. If it was a pattern that was interesting, one must look at the space of all interesting patterns and look at the probability that one might be found. This is the difference between math and numerology, or astrology and astronomy. Coincidence has a role, why it should even be expected; for christs sake it is even predictable. Perhaps some of the people reading this are aware of the fact that if you get about thirty people or so, together two of them are likely to share a birthday. This does not mean that if you pick any person that someone else is likely to share their birthday. Rather of the group there is a good chance two of the individuals, whomever they may be, will likely share the birthday. If you pick 42424242 before hand, that makes it unlikely, if you don't it makes it coincidence.
--Jimmy has fancy plans; and pants to match.
In statistics classes, most (that have been there) have heard the idea: An infinite number of monkeys can type a whole Sheakspearean piece... It sounds like we don't need all those monkeys, and can find all books, past/present and future in PI!!!!!
In other news today, the RIAA has sued mathematicians everywhere under the DMCA for circumventing pi.
The Hotel in question is called a Hilbert`s hotel, named after the mathematician who created the concept. For a completely mindblowing read concerning all things infinite aleph 0.. aleph 1 etc, check out Rudy Rucker "White light" - it will blow your mind!
A crash reduces
Your expensive computer
A crash reduces
Your expensive computer
to a simple stone.
not exactly - pi will contain all information in all bases. any number in base 8 (for example) can be represented by a number in base 10 which you can find in pi as normal. Looking in another base my help you to find a message quicker (or it may not, there is no way of knowing).
:-)
as an aside, I was reading this yesterday and the thought struck me that carl sagan was correct, there is a bitmap of a circle, written in base 11 (or any other base), conatined in pi. Anybody fancy encoding that bitmap in base 11, finding it in pi, and posting the results on a number of sci-fi fansites - just to blow their minds
A crash reduces
Your expensive computer
A crash reduces
Your expensive computer
to a simple stone.
Don't be ridiculous. Why do you think the start end end indexes into pi will be of shorter length than the message itself?
Then you were taught wrong. In fact I can prove the opposite. If PI reccurs within PI in any number base then PI is a rational number, therefore it cannot reccur.
The reason is that if we have 0.123123123... then we can divide it by 1000 and subtract it from itself to get 0.123, so if x - 0.001 x = 0.123 then x = 0.123 / 0.999 => x = 123/999
The proof can be trivially extended to cover numbers of the form 0.1111123123123123 ...
The flaw in the original logic is that a proof that every finite string occurs in PI does not consititute a proof that every infinite string occurs in PI.
Looking for an Information Security student project suggestion?
Try http://dotcrimeManifesto.com/
LOL!! I'd mod you up for your sig if I could. :)
While I think your sidenote is interesting, I think there is a flaw in it. If you watch a game of basketball and a player is "in the zone" you will often see the opposition react to this and they will add more defensive pressure to the player. Thus, for the player to continue being streaky they actually have to work harder to score, thus indicating that if they continue to do so, they are probably in an actual "zone" and that it isn't just a case of simple statistics.
If it's supposed to move and doesn't, use WD-40. If it moves and it shouldn't, use duct tape.
Take in example the question what the squareroot of -1 yields? Maybe, "who cares"??? When people started to think about imaginary numbers many people thought that, but today in electric calculations with AC current you can't even imagine how to handle it without the imaginary i.
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Karma 50, and all I got was this lousy T-Shirt.
"Information wants to be pi"
-josh, who needs to do something better with his time
Here we go again, Fermat!
(Pardon me for thinking you already had your last theorem!)
infinitely large numerator and denominator? :)
That probably would, almost by definition, make it an irrational number.
i'd normally be the one complaining about poorly worded and misrepresented math articles, but you did pretty well on that one.
much appreciated.
A: None. The Universe spins the bulb, and the Zen master merely stays out of the way.
If pi has all conceivable messages, pi must contain all of the US military's secrets, DeCSS, kiddie pr0n, violent and explicit sexual films beyond anyone's imagination and much much more. It must therefore be banned. When you get the death penalty for circle possession, don't say I didn't warn you...
Hmm... Check this out.
Check this formula too. If the formula is so simple (like taking the sigma out of a mere factors of fractions), pi couldn't be containing any message. I simply skeptic about that....
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Error 500: Internal sig error
Why not chop off two of each researcher's fingers, so that they can use base 8 instead of base 10? I hear it's much easier to calculate base 8 digits of Pi than base 10. Surely there must be a theorem somewhere stating that normality is independant of the base. Is it?
Music speeds up when you yawn, but does not change pitch.
The philosopher Lakatos did some thinking about how if you're working with a definition, and then some object that you thought should fit your definition turns out not to, should you change your definition to include the object, or should you exclude the object?
He reasoned (correctly) that either course of action is viable. It's an arbitrary decision as to which you choose.
So, whenever you hear someone arguing about whether 1 is prime or other such arguments, simply yell "Semantics!" and end the conversation.
Music speeds up when you yawn, but does not change pitch.
Strictly speaking, the property mentioned isn't actually normality, but normality to base 10^n for all n. Normality to base b means that if you write down the base-b expression for the number then every base-b digit occurs with equal frequency. So normality to base 10 means that in the usual decimal expansion, 3 and 7 occur with equal frequency, for instance. Normality to base 100 means that, e.g., in the decimal expansion 34 and 87 occur with equal frequency.
It's known that in a certain precise sense, almost all numbers are normal (i.e. normal to *all* bases). But to this day, not one single specific number has been *proved* to be normal!
Well we are close to finding the true name of GOD according to the Hebrews and ALso there should be a predictability algoith for the stock market as well correct. They need to make another TT movie TT sqared: Tracing the origins of the Red Code virus back to TT.
"All I can tell the "lesser of two evils" folks is that if they keep voting for evil, they'll keep getting evil."-Lp.org
Not necessarily. Consider pi, written base 10. Now replace every occurrence of the digit 7 with the digit 2. The resulting number is still infinitely long, non-repeating, and random (but I suppose this depends on what your definition of random is). It doesn't contain any finite number that contains a 7 and thus doesn't contain all finite numbers.
WHO CARES!
OK so the ratio of the circumference of a circle to it's diameter just happens to be a number that has infinite decimal places and contains an equal distribution of all possible base 10 numbers... what in the hell does this prove? First of all, it's not proven anyway, and second of all, do these people think that Pi is the secret to life or something? Really folks, it's just a number, and since we have no practical means of measuring anything past the... oh... say... 1,000,000,000ths digit, why bother?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ now you know
Gently, while pondering the problem du jour, scribbles meaningless symbols on a piece of paper, proclaiming that he's now reduced a seemingly impossible mystery into a relatively simple problem of cryptography...
I think that this is very similar, if memory serves, to what was described in the movie PI.
Is it possiable that the numbers that are extracted from the irrational number have some kind of signifigant meaning in nature?
The key to GUT'S in a piece of PI??
~Shane
Whatever....you post this same shit all the time. Find something better to do with your time you fuckin _troll_
A specific string of k bits long will on average occur every 2^k bits of a random bitstream. (For example, it will on average take about 1024 coin tosses to get 10 heads in a row.)[0] The reasoning I have in mind for this relies on the fact that coin tosses are independent events. The digits of pi are certainly not independent random events because we can calculate them. However, because of normality behaves macroscopically much like independent events, k precise bits will occur on average every 2^k bits of pi. I imagine the actual proof is highly nontrivial, but handwavy entropy arguments convince me that this is true.
.320 and they
play 1000 games (made-up numbers).
[0] As an interesting sidenote, many of the "streaks" in sports coincide roughly with the streaks that probability dictates. Baseball hitting streaks aren't necessarily because the hitter is in the zone... they may be because probabalistically, they are due to hit in 30 games in a row if their average is
couldn't you in theory find a specific position where pi prints out say, the Max Payne ISO, and distribute that position to friends?
I've been thinking about this. As many others have pointed out, the index would take as much space as the original data.
However, suppose you can generate a short mathematical expression that will put you in the nieghborhood of the index. Let's call this point A.
Now, find point B close to the index (perhaps 1000 digits) such that it contains a string, S, which does not occur between point A and point B-1. Now all you have to do, is note the offset from point B and the length of the data.
To decompress, you compute the expression for point A, search pi starting at A, and stopping when you find S. You are now at point B. Compute the offset from B and generate your data.
So, if you could get your expression for point A down to a few bytes, and your string for point B down to a few bytes, you would indeed have incredible compression. Of course, the search for S could take almost as long as the compressing the data.
To improve upon this idea, instead of using pi, what if you could find a function with the following properties:
Searching pi until you find that right position that matches your Max Payne ISO, which could be located on the far end of infinity.
Good luck!
Douglas Adams, Hitchhiker's Guide
healyourchurchwebsite.com - WWJB?
If PI contains every concievable message, then do you suppose we could use to predict the future ? Not the normal boring stuff, like will my daughter run off with tall swarthy mediterranean, but important stuff, like assasinations ?
healyourchurchwebsite.com - WWJB?
healyourchurchwebsite.com - WWJB?
(Don't laugh at me! I can still kick your butt in chemistry!)
I recall a method for detecting falsified data that relied on the fact that certain functions (those that followed a log pattern, I think) produce results with the last digit's frequency skewed toward smaller numbers (1, 2, 3,...) and AWAY from larger numbers (9, 8, 7,...).
So my question is, does Normalcy apply to functions (in the way I described), as well as numbers?
The digits of PI are relatively random where 1.2112211122211112222.... have a relatively predictable pattern in decimal. I know PI is not a random number but it's decimal places are random and in the long run should have equal displacement of each number.
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Just because a bunch of people believe or do something stupid, doesn't make it any less stupid.
My phone number (minus area code) starts at digit 18740826 (not including the 3) of pi.
D/\ Gooberguy
Karma: Meh (Mostly from meh.)
If Pi is infinitely long, non-repeating and random, then isn't the rule that any such number must contain all finite numbers ... eventually?
But, by extrapolation, does this not imply that while pi may contain the complete source code to Office 2000, it also contains all possible incorrect versions, and it is impossible to know which one you have found. And it is impossible to know which base to look in. And it is impossible to know how far in it is. And, and, and,...
Essentially, if you find anything meaningful in pi (or e, or ...), then it is a: accidental, and b: not actually meaningful.
The mathematics of trans-finite numbers will make your brain melt if you think about them long enough.
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I see no THERMONUCLEAR WARHEAD here.
You are at Y2.
"This is a Hollywood movie: when it comes to the Laws of Physics, they're lucky if they get Gravity!" --- my wife
The fact that they are looking for patterns is kinda cool. Anyone else think that they should compare it with pattern's in Chaitin's Omoga. Would be nice, if you could calculate it though. Unfortunately, knowing a single digit of that number without calculating its compononts is a paradox, sigh.
badness 10000
Does anyone have a search program running that'll search throughout Pi?
If so, i wonder where in Pi the following numbers would occur:
072101108108111044032087111114114108100046
=
Hello, World.
Segmentation fault(core dumped)
"The new work arises out of a surprising formula for pi that Bailey and his colleagues discovered in 1996, which allows mathematicians to compute any digit of pi without knowing the previous digits" Yet appeared on /. less than a week ago. Latency in news is yummy.
Also, this could be of use in generating keys for other encryption schemes.
It may be true that pi contains every possible sequence of digits. So what? To get a message I have to use a code. But when I can choose the code as I like, I can get every message out of any fixed string of digits.
It may be more obvious for you that 1337 means "elite" than "1" meaning the DeCSS-sourcecode, "2" meaning the 2.4.8-source, and so on. But ask someone on the street what 1337 means. You will see that it is not the number that contains the information, but it is your code. "1337" is only the trigger.
Disclaimer: Since I read GEB I'm confused. Not because of the things I didn't understood, but because of the things I understood.
Hmmm, so the alien message must be the one string not present in pi. Let me just do a few quick calculations here...
clickity clickity click
Ah, finally a message from the creators of the universe: WE DO CHICKEN RIGHT
What the...?
Magnatune: Quality (DRM-free) MP3/FLAC/
The second task was to add an infinite number of guests to the hotel... Solution: Tell every current guest to move to the room number double his/her current room. Tell the new guests to go to the odd numbered rooms.
Sorry, I don't where this story was first published...
Funny. My mom can never seem to slice pi with the same kind of normality. Then again, my mom is as odd as pi anyway...
Blog Prophyts - Right On, Man
If Pi is pretty much random (aside from the 3.14), then what the hell happens to pi(r)? Does this mean that a circle has a different circumferance each time? Thank the maker I'm a designer and not a number cruncher. And does anyone know the equasion they used to get pi from the base of the Pyramids of Giza? I remember seeing it like 10 years ago on TV...
Blog Prophyts - Right On, Man
The fact that you can encrypt numbers with a set of random digits by adding your message to the random number is pretty standard - as far as I know, it's the only unbreakable code. You just use a different sequence of random numbers each time. Create two copies of the sequence, give one to the person you will want to eventually decrypt your message, and you're golden. Adding the offset-from-pi thing makes the code WAY less secure. Let's say you start your message with the offset from pi: 3746 2944 2345 or whatever, with the first four numbers being the offset, say. Now all your enemy has to do is figure out that you're using pi, and you're done for. So that's not an unbreakable code at all. An offset from pi is, after all, not a random number, it's a known one. Incidentally, this is sort of like that AOL thing a couple of days ago - if you want to maintain connection with our servers, you have to send us a message encrypted from a specific offset off of the aol.exe file! So to connect with AOL at all, you need AOL's executable. But, of course, an offset off of PI is easier to come by...
I recently placed two conductive wires into the water in my toilet. I connected the wires to a Fluke multimeter and then passed the collected voltage data to my PC. Don't waste time with monkeys and typewriters or analysis of PI, people! I found a detailed description of how to create controled nuclear fusion embedded in my collected data! Don't believe me? Hey, would I lie?
Yeah, but what's the point of finding, "First Post!", in pi if it's not first?
Okay... I really know very little about this kind of math. But it seems to me that we are talking about an irrational number with infinite digits. Just keeping with base ten for a second, does that mean that there are the exact same number of 1s, 2s, 3s, etc. (strings of length one)?
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Carl Sagan addressed the issue of a message in pi in the book Contact:
Occasionally you'll get a few consecutive digits that are the same--4444, for example-- but not more than you'd expect statistically. Now, suppose you're running merrily through these digits and suddenly you find nothing but fours. Hundreds of fours all in a row. That couldn't carry any information, but it also couldn't be a statistical fluke. You could calculate the digits in pi for the age of the universe and, if the digits are random, you'd never go deep enough to get a hundred consecutive fours."
"You mean you could decode a picture hiding in pi and it would be a mess of Hebrew letters?"
"Sure. Big blade letters, carved in stone." He looked at her quizzically.
So, if pi contains every base 10 string of length n at offset x, at what point do the strings become longer than the offsets? In other words, if I was searching for the base 10 representation of the 9000 names of God (a fairly long string) what is the possibility that I could represent it as an offset within Pi and save space? If nothing else, this seems like a fairly neat method of encryption, but seems to have very little practical application. Eh, someone probably already said all of this. In fact, this message can be found at Pi offset 386512308951618357012834565.
Happiness is relative, Based upon the way we live.
I just realized, this is a completely backwards approach to the whole thing. Sure, pi is easy to find in nature all around you, circles are fairly common. But unless the value you're looking for is 314159... you're stuck with more digits than you started with. What we really need is a method of finding a calculable irrational number for any value. According to most of the more egotistical mathematicians I've met, "Anything can be represented as a formula." At that point, all hell breaks loose, though. Your Max Payne ISO number (The Payne Constant) has no mathematical signifigance at all other than being a complete representation of a popular video game. Huzzah for you and your number, but good gosh. The Payne Constant formula is being posted to newsgroups, bulletin boards, heck, there's a car over there which has it on a bumper sticker. That guy over there is wearing a t-shirt with it printed on it. It's on my business cards, and I end all my phone calls by reciting it. Bang goes software development of any kind, and in marches copyright law reform. Then again, that might be a good thing... And from there it gets worse. Why stop at a single formula for Max Payne? How about the complete Playstation ISO collection, as a single number? The Absolute Library of Congress (updated yearly) as a single number? _Insert interesting archive here_ as a single number? It reaches the point where, even if such a program for calculating formulae for irrational nubmers _could_ be produced, that doesn't mean it _should_ be produced. Then again, if it _can_ be produced, human nature dictates that it probably _will_ be produced. There's an obsession for every person, and a person for every obsession. Everything that is possible will be done. But now I'm reaching into the realms of philosophy, and dodgy philosophy at that. All in all, this is probably far more effort put into a post which will never be read than is rational.
Happiness is relative, Based upon the way we live.
As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments:
1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator.
2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi.
3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n).
4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg
5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site: http://www.nersc.gov/~dhbailey
Cheers, David H Bailey david@dhbailey.com
As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments: 1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator. 2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi. 3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n). 4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg 5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site: http://www.nersc.gov/~dhbailey Cheers, David H Bailey david@dhbailey.com
The infinite hotel is a standard problem in Number Theory, named after some famous Math geek whose name (conveniently) escapes me. But what about appending an infinite number of infinite numbers together? That is, what if the above scenario (aleph-null) were to be repeated an infinite number of times (aleph one)? All goes back to that other famous theory named after another forgettable math genius, that any space or surace contains an infinite number of mathematically measureable points (cartesian), regardless of it's size. Thinking here that it's interesting that human beings as finite as we, are able even to conceive of infinity. Imagine what we could conceive of if we were infinite. But that's philosophy. I'll go find another thread.
I am alone, yet I also surf the universal backwash of undifferentiated Being, which is LOVE.