h a l f b a k e r yI CAN HAZ CROISSANTZ?
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When Bricks of Pi are being
manufactured, each block receives a four
digit
number stamped onto its outside face,
prior to firing.
These numerals are extracted out of the
infinite series that results from the
evaluation of Pi. You can lay them in any
order, or request a precise sequence
corresponding to the number of bricks
required to construct your entire
building.
Also available - The House That Genome
Built.
pi to 50,000 decimal places
http://www.geocitie.../1216/numtab/pi.htm
or is it an outside wall of that new science tower? [xenzag, Jul 20 2008]
Normal Numbers
http://en.wikipedia.../wiki/Normal_number
In mathematics, a normal number is, roughly speaking, a real number whose digits (in every base) show a uniform distribution... [futurebird, Jul 20 2008]
every conceivable sequence....
http://www.inwit.co...ndodditiesofpi.html
occurs in pi [xenzag, Jul 21 2008]
Search pi for strings
http://www.angio.net/pi/bigpi.cgi
My SS number doesn't occur in the first 200 million digits, but a string of the first 8 digits does, and also the strings 00000000, 11111111, 22222222, 121212121, and 12345678. [ldischler, Jul 21 2008]
The Pi Code
http://users.aol.com/s6sj7gt/picode.htm
[ldischler, Jul 21 2008]
Today is Pi approximation day
http://en.wikipedia.org/wiki/Pi_Day
[DenholmRicshaw, Jul 22 2008]
[link]
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//You can lay them in any order//
Of course you can, and it's still a sequence of pi. |
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it has been proven that any arbitrary sequence of numbers appears somewhere in the sequence of pi. |
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(I realize that this is widely believed, but I find it difficult to prove - it's a different statement from "there are infinitely many digits in the decimal expansion of pi" or from "pi is non-repeating".) |
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I just went to that "pi to 50,000 decimal places" link, opened my find-in-this-page editor, and typed in random sequences of numbers. I got most 5-digit sequences, and a few 8-digit sequences. |
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"Any arbitrary sequence of numbers" would mean we could somewhere find the digital equivalent of _Hamlet_--that's too much to expect. You'll need to get your house in order. |
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Careful, [snictown], the word "proof" in mathematics can be a bit tricky. |
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The idea about Hamlet is true if pi is a normal number, and most people think it is normal. |
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It seems that given the infinite nature of
the pi sequence, then [snictown] could
be right - this is a quote from the link.
"Now, if this be true, that p is
completely random, then every
conceivable sequence of numbers must
somewhere occur in p . That is, if one
considers any number, 12345678 for
instance, that number will sooner or
later be found in p if the search is
pressed far enough. To assume
otherwise would be to prejudice the
randomness of p." |
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First, I don't know of any proof that pi is a random sequence of digits. I'm aware that lengthy sequences have been tested and found to be random, but that's different from being able to assert that the entire, infinite, sequence of digits in pi follows a completely random distribution (or are there, for example a fraction of a percentage more sevens in the sequence than you'd expect?).
Second, it seems common sense to say that "if the digits in pi follow a random distribution then every conceivable sequence of numbers must somewhere occur in pi" but I'm not sure that's actually true. To put it another way, if pi is random, then does that 'prove' that "1234567890" occurs somewhere in pi, or does it just make it very probable? |
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Either way, I'm not that concerned. If I
understand
Gödel's incompleteness theorem well
enough, it
leaves enough room for no proof or
perhaps of more
importance, no disproof. As Pi DOES
have a verifiable
sequence, then I see no problem. |
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//it has been proven that any arbitrary sequence of numbers appears somewhere in the sequence of pi.// |
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[sninctown] It is conjectured, not proven. |
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//Second, it seems common sense to say that "if the digits in pi follow a random distribution then every conceivable sequence of numbers must somewhere occur in pi" but I'm not sure that's actually true.// |
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[hippo], if pi were to be proved a normal number, it would be a disjunctive sequence, and hence every finite string would be represented. |
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[baconbrain] If it were a normal number, it would show uniform distribution in all bases. So Shakespeare, as well as Beethoven, and everything else (converted to binary), would have a representation in the binary representation of pi. As would this anno. |
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However, we are a long way from finding pi to be a normal number. |
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[HegelStone] Not true, discussed before on the 'bakery, but it makes a great tagline! |
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//So Shakespeare, as well as Beethoven, and everything else (converted to binary), would have a representation in the binary representation of pi.//
Even messages could be found there, and will, with enough computing power. Hidden messages from God himself. And one day we’ll see a book titled, “The Pi Code: Finding God in Transcendental Numbers.” |
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Safe in the knowledge, that every ELS, would appear as plaintext somewhere hence, so you only have to expand pi to maximum half its actual places:-). |
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If you make certain of the bricks removable, then when the mood strikes you you could switch them around. |
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You could then invite people to come to your house, show them around, and when they say, "This is the same place you've been for the last five years," you could provably respond, "No, it's just a similar house in the same geographical location. Now give me my housewarming gifts." |
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Repeat until you have a full china set and no friends. |
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The key to the answer to the question being asked is not so much the randomness of pi, but the infinite-ness of its sequence of digits. It is in that infinity that one could expect to find any FINITE arbitrary sequence. I conjecture it would be possible to find a place, somewhere in that infinity, that begins, 314159265358979323846... and proceeds to duplicate all the initial digits of pi, until this particular starting point is reached. After the duplication, effective randomness would follow, for a nice long stretch, and eventually a NEW starting point could be found, which duplicates everything up to THAT point. And so on.... If you don't think this is possible, then what it really means is that you don't properly understand what "infinity" allows. |
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No, Vernon, infinity doesn't imply normalness. |
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Here's how to build a counterexample: Take PI, translate it into binary, then read those digits back as decimal digits. The resulting number goes something like this: "11.0010010000111111..." That gets you another series of decimal digits that has roughly the same properties of infinity and non-repeatingness as PI -- except that it doesn't contain the digits 2 through 9. (It is thus not a normal number.) That sequence is still infinite, but doesn't contain the substring "2". |
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I agree that it's likely that PI is a normal number, but it's interesting that it's not (yet?) proven. |
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[jutta], you appear to be mixing conditions. I was not talking about a binary form of pi. But if I was, I would simply say that you ought to be able to take some arbitrary length of the initial ones and zeros (instead of some arbitrary length of Base-10 digits), and eventually find a duplicate of them, somewhere in the infinite sequence of ones and zeros. |
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A piece of trivia for those who haven't seen it before: pi can be computed as the sum of the infinite series 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 ... (don't try this without a computer; it takes a LOT of terms for the result to gradually converge into even the first few digits of pi) |
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I'm not talking about the binary form of PI, I'm just using it to build a decimal number that is just as infinite as PI, yet doesn't contain some sequences of digits. That is a counterexample to your claim that infinity implies normalness. |
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// Repeat until you have a full china set and no friends.
[marked-for-tagline] |
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[jutta], OK, I know there are contrived infinite sequences with properties such that you describe. Pi has not been shown to be such a sequence, just as it has not been shown to be random enough for "normality". I think, though, that it is random enough for most modest/finite arbitrary sequences of digits that one might care to specify, those sequences can be found somewhere among the digits of pi. I did say in my first post here that the key was less a matter of its randomess than a matter of its infinite-ness, and in saying that I did not mean to imply that the degree of randomness which it exhibits is discount-able. |
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I am compelled to quote from Gödel (via
wikipedia)
"Gödel's first
incompleteness theorem
shows that any formal system that
includes enough of
the theory of the natural numbers is
incomplete: it
contains statements that are neither
provably true nor
provably false. Or one might say, no
formal system
which aims to define the natural
numbers can actually
do so, as there will be true number-
theoretical
statements which that system cannot
prove." |
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This is all a bit moot. [ldischler] said "Of course you can"; [jutta] said "?"; [snictown] said "yes. It's proven."; [jutta] said "I don't think so". |
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Since then, the cut and thrust of the idea has been sidelined and there has been a debate about the randomness of Pi. Pi is not a random number, although it's irrational sequence appears to be so, and it is known to over a trillion decimal places. |
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Therefore, I suggest that it does not matter whether the 42,000 combinations of brick-numbers form a section of Pi. They will look like it - and no-one will be able to realistically mount a counter-proof. |
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<I'm reminded of kids in a playround: "Your Mum *smells*", "No she doesn't", "Yes she does", "Prove it then", "No. You prove she doesn't smell"...> |
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Assuming that the infinity/normallness thing does apply to pi, then it may also apply to other numbers of the same ilk (pi+1, e, e-9, e+pi, phi, among others) In which case any sequence of arbitrary numbers must 'belong' to any number of arbitrary transcendental numbers. |
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So you have an arbitrary sequence of bricks which *may* be found in any of an infinite and uncountable number of transcendental numbers - am I alone in thinking "Big Deal"? |
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Anyway, if you want a house that's piesque, shouldn't it be circular? |
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At least the bricks aren't representations of sqrt(-1) - Who would want to live in an imaginary house? |
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I dunno, it seems like a lovely place (in my head). |
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To be fair, it is a nice house, it's just the gaping enlegged coelacanth lurking by the back door that gives me the heebeegeebees. |
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If pi contained *every* possible sequence of numbers within its digits, then we would have to conclude that somewhere within pi is a never-ending string of 8's, just as long as pi itself. |
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//If pi contained *every* possible sequence of numbers within its digits, then we would have to conclude that somewhere within pi is a never-ending string of 8's, just as long as pi itself.//
Every possible *finite* sequence. |
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I read every word and now my head hurts.
Would there be infinite strings of prime numbers scattered throughout Pi then?
<goes for an Aspirin> |
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[2 fries shy of a happy meal]:
Short answer: Yes
Long answer: <Morpheus> Unfortunately, no-one can be told how infinite Pi is - you have to prove it for yourself. </Morpheus> |
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Hmm.
By analyzing reoccurring sequences of prime number sequences within Pi would it possibly make it easier to... |
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...nope...I lost it. My bad. Carry on. |
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Anyone else think of Heinlein's "--and he built a crooked house" when the imaginary house came up? I know a tessarect isn't imaginary, but it should be mentioned somewhere in a discussion of mathematical houses. |
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//Anyway, if you want a house that's piesque, shouldn't it be circular?// pi is a *ratio* that *crops* up everywhere, not just in circles. Even in your imaginary house. In fact, specifically, in your imaginary house. |
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Indeed. It is dismantling to be satisfied with the fact that a simple Leibniz series yeilds the digits of pi: |
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4/1 - 4/3 + 4/5 - 4/7 ... = pi |
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Whenever I see a series with x1 - x2 + x3 - x4 + ... |
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I think i, pi, i ?!?! :-) |
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Oddly enough, whenever I see a system that begins x1+x2+x3....
I think, "I'd like some pie." |
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So next time someone asks me for my
number I just write the symbol for 'pi'. |
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I wonder how many people have those first few digits for their pi-n number. |
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In an undergraduate lecture, a tutor put the numbers 1,4,1,5,9 up on the board. He said, "I have a shiny new textbook for the first student who can tell me the next two numbers in this sequence..." |
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All the students hastily got their notepads out and started to calculate whatever. All the students except [Jinbish] - he'd recognised the sequence straight away. A friend of his had memorised Pi at high school. [Jb] had been sitting next to him as he recited out loud 3.1415926 over and over again. [Jb] had almost fallen out with his repetitive mate... but now fate had smiled on his near madness-inducing episode. |
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"2 and 6!" and as the tutor looked up and smiled, and a hundred astonished faces turned round, he knew that both the textbook and infinite amounts of derision from the rest of the student body were now his!! |
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