7 ms·

As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. So as we have so many piece

As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. So as we have so many pieces of pi now, we could create a file sharing system that's not based on sharing the data, but the position of a piece of the file in pi. That would be kinda funny

sammex 2y ago
Would the index number actually be smaller than the actual data?
- psychoslave 2y ago
  You need both index and length, I guess. If concatenating both value is not enough to gain sufficient size shrink, you can always prefix a "number of times still needed to recursively de-index (repeat,start-point-index,size) concatenated triplets", and repeat until you match a desired size or lower. I don’t know if there would be any logical issue with this approach. The only logistical difficulty I can figure out is computing enough decimals and search the pattern in it, but I guess that such a voluminous pre-computed approximation can greatly help.
  - waldrews 2y ago
    No invertible function can map every non-negative integer to a lower or equal non-negative integer (no perfect compression), but you can have functions that compress everything we care about at the cost of increasing the size of things we don't care about. So the recursive de-indexing strategy has to sometimes fail and increase the cost (once you account for storing the prefix).
    - psychoslave 2y ago
      Is there some inductive proof of that? Or is that some conjuncture? Actually any resources related to that point could be fun to explore
      - waldrews 2y ago
        It's a classic application of the pigeonhole principle, the first on in this list: https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_applications https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_...
- waldrews 2y ago
  It would average the same size as the actual data. Treating the pi bit sequence as random bits, and ignoring overlap effects, the probability that a given n bit sequence is the one you want is 1/2^n, so you need to try on average 2^n sequences to find the one you want, so the index to find it is typically of length n, up to some second order effects having to do with expectation of a log not being the log of an expectation.
Moosturm 2y ago
https://github.com/philipl/pifs https://github.com/philipl/pifs
maxmouchet 2y ago
https://news.ycombinator.com/item?id=8018818 https://news.ycombinator.com/item?id=8018818 and https://github.com/philipl/pifs https://github.com/philipl/pifs :-)
IsTom 2y ago
There are many ways in which a number might not never repeat itself, but not contain all sequences (e.g. never use a specific digit). What you want is normal numbers and pi is not proven to be one (though probably it is).
voytec 2y ago
> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. You reminded me of this Person of Interest clip: https://www.youtube.com/watch?v=fXTRcsxG7IQ https://www.youtube.com/watch?v=fXTRcsxG7IQ
A_D_E_P_T 2y ago
> every piece of conceivable information (music, movies, texts) is in there, encoded Borges wrote a famous short story, “The Library of Babel,” about a library where: “... each book contains four hundred ten pages; each page, forty lines; each line, approximately eighty black letters. There are also letters on the front cover of each book; these letters neither indicate nor prefigure what the pages inside will say. “There are twenty-five orthographic symbols. That discovery enabled mankind, three hundred years ago, to formulate a general theory of the Library and thereby satisfactorily resolve the riddle that no conjecture had been able to divine—the formless and chaotic nature of virtually all books. . . “Some five hundred years ago, the chief of one of the upper hexagons came across a book as jumbled as all the others, but containing almost two pages of homogeneous lines. He showed his find to a traveling decipherer, who told him the lines were written in Portuguese; others said it was Yiddish. Within the century experts had determined what the language actually was: a Samoyed-Lithuanian dialect of Guaraní, with inflections from classical Arabic. The content was also determined: the rudiments of combinatory analysis, illustrated with examples of endlessly repeating variations. These examples allowed a librarian of genius to discover the fundamental law of the Library. This philosopher observed that all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet. He also posited a fact which all travelers have since confirmed: In all the Library, there are no two identical books. From those incontrovertible premises, the librarian deduced that the Library is “total”—perfect, complete, and whole—and that its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite)—that is, all that is able to be expressed, in every language.” I've done the (simple) math on this -- in fact I'm writing a short book on the philosophy of mathematics where it's of passing importance -- and the library contains some 26^1312000 books, which makes 202T look like a very small number. So though everything you describe is encoded in Pi (assuming Pi is infinite and normal) we're a long, long way away from having useful things encoded therein... Also, an infinite and normal Pi absolutely repeats itself, and in fact repeats itself infinitely many times.
- WillAdams 2y ago
  And for an amusing example of this see: https://www.piday.org/find-birthday-in-pi/ https://www.piday.org/find-birthday-in-pi/
NooneAtAll3 2y ago
> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. may I interest you in the difference between *irrational* numbers and *normal* numbers? look at https://en.wikipedia.org/wiki/Liouville_number https://en.wikipedia.org/wiki/Liouville_number - no repeats, but minuscule "contained information"
euroderf 2y ago
> every piece of conceivable information (music, movies, texts) is in there, encoded. So that means that if we give a roomful of infinite monkeys an infinite number of hand-cranked calculators and an infinite amount of time, they will, as they calculate an infinite number of digits of pi, also reproduce the complete works of Shakespeare et al.
- _joel 2y ago
  and then do it all again, but backwards.
mkl 2y ago
> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. This is true for normal numbers [1], but is definitely not true for all non-repeating (irrational) numbers. Pi has not been proven to be normal. There are many non-repeating numbers that are not normal, for example 0.101001000100001... Storing the index into pi for a file would usually take something like as much space as just storing the file, and storing or calculating enough digits to use that index would be impossible with the technology of today (or even probably the next century). [1] https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number
- tombert 2y ago
  It's conjectured to be normal isn't it? I know it hasn't been proven yet, and I cannot seem to find where I read this, but I thought there was at least statistical evidence indicating that it's probably normal.
  - adgjlsfhk1 2y ago
    100% of real numbers are normal, so that's pretty strong statistical evidence
    - hn_throwaway_99 2y ago
      What? No they're not, e.g. no rational numbers are normal, and they are real.
      - GraphEnthusiast 2y ago
        The rational numbers make up "zero percent" of the real numbers. It's a little hard to properly explain without assuming a degree in math, since the proper way to treat this requires measure theoretic probability (formally, the rationals have measure zero in the reals for the "standard" measure). The short version is that the size of the reals is a "bigger infinity" than the size of the rationals, so they effectively have 'zero weight'. Reference (very technical): https://math.stackexchange.com/questions/508217/showing-that-rationals-have-lebesgue-measure-zero#508225 https://math.stackexchange.com/questions/508217/showing-that...
sxv 2y ago
Isn't 202TB (for comparison) way too small to contain every permutation of information? That filesize wouldn't even be able to store a film enthusiast's collection?
- RamblingCTO 2y ago
  Well it all comes down to encoding, doesn't it. We can represent almost everything with just 0 and 1 as well, can't we? The description of that data is way bigger than the elements used to describe it of course.
_fizz_buzz_ 2y ago
This is not necessarily true. Pi might not repeat but it could at some point - for example - not contain the digit 3 anymore (or something like that). It would never repeat, but still not have all conceivable information.
- pilaf 2y ago
  But the number 3 is there just because we decide to calculate digits in base 10. We could encode Pi in binary instead, and since it doesn't repeat it necessarily will never be a point where there will never be another 1 or a 0, right?
  - bubblyworld 2y ago
    That's true - you can quite easily prove that an eventually constant sequence of decimals codes for a rational number. But it's also true that pi may not contain every _possible_ sequence of decimals, no matter what base you pick. Like the Riemann hypothesis, it seems very likely and people have checked a lot of statistics, but nobody has proven it beyond a (mathematical) shadow of doubt.
  - _fizz_buzz_ 2y ago
    Obviously, it was just an example to illustrate what a non-periodic number could look like that doesn’t contain all possible permutations. If the number never contains the digit 3 in base 10 it will also not contain all possible permutations in all other bases.
    - deleted 2y ago
      [deleted]
criddell 2y ago
> every piece of conceivable information is in there Wouldn't the encoded information have to have a finite length? For example, pi doesn't contain e, does it?
- tzs 2y ago
  > For example, pi doesn't contain e, does it? Assuming we are only interested in base 10 and that pi contains e means that at some point in the sequence of decimal digits of pi (3, 1, 4, 1, 5, 9, 2, ...) there is the sequence of decimal digits of e (2, 7, 1, 8, 2, 8, ...), then I believe that question is currently unanswered. Pi would contain e if and only if there are positive integers n and m such that 10^n pi - m = e, or equivalently 10^n pi - e = m. We generally don't know if combinations of e and pi of the form a pi + b e where a and b are algebraic are rational or not. Even the simple pi + e is beyond current mathematics. All we've got there is that at least one of pi + e and pi e must be irrational. We know that because both pi and e are zeros of the polynomial (x-pi)(x-e) = x^2 - (pi+e)x + pi e. If both pi+e and pi e were rational then that polynomial would have rational coefficients, and the roots of a non-zero polynomial with rational coefficients are algebraic (that is in fact the definition of an algebraic number) and both pi and e are known to not be algebraic.
- RamblingCTO 2y ago
  I implied that, yes
worewood 2y ago
The sad thing is that the index would take just as much space as the data itself, because in average you can expect to find a n-bit string at the 2^n position.
Zambyte 2y ago
https://news.ycombinator.com/item?id=36357466 https://news.ycombinator.com/item?id=36357466
2OEH8eoCRo0 2y ago
Does pi contain pi?
- schoen 2y ago
  It does, starting right at the beginning!
sundry_gecko 2y ago
Reminds me of a scene of Finch teaching in Person of Interest. https://m.youtube.com/watch?v=yGmYCfWyVAM https://m.youtube.com/watch?v=yGmYCfWyVAM
constantcrying 2y ago
>As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. It is somewhat shocking that again and again this logical fallacy comes up. Why do people think that this is true? It doesn't even sound true.
- hkhanna 2y ago
  Isn't it a property of infinity? If pi goes on infinitely without repeating itself, every possible combination of numbers appears somewhere in pi. It's sort of like the idea that if the universe is infinitely big and mass and energy are randomly distributed throughout the universe, then an exact copy of you on an exact copy of Earth is out there somewhere. This property of infinity has always fascinated me, so I'm very curious for where the logical fallacy might be.
  - n2d4 2y ago
    Not necessarily. The number 1.01001000100001000001... never repeats itself, yet most other numbers can never be found in it. A number that contains all other numbers infinitely many times (uniformly) would be called normal, but no one has managed to prove this for pi yet. In fact, no one even managed to prove that pi doesn't contain only 0s and 1s like the above after the X-th digit.
  - constantcrying 2y ago
    >Isn't it a property of infinity? If pi goes on infinitely without repeating itself, every possible combination of numbers appears somewhere in pi. No. Example: 0.1011011101111011111... does never repeat, yet there is no 2 in there, neither is there 00 in there.
    - onion2k 2y ago
      The fact you can't encode arbitrary data in a structured-but-irrational number doesn't mean you can't encode data in a 'random' irrational number. The question is really 'Does every series of numbers of arbitrary finite length appear in pi?' I can't answer that because I'm not a mathematician, but I also can't dismiss it, because I'm not a mathematician. It sounds like a fair question to me.
its_ethan 2y ago
https://libraryofbabel.info/ https://libraryofbabel.info/ you might find this to be pretty cool. It's similar to what you're describing. Whoever made it has an algorithm where you can look up "real" strings of text and it'll show you where in the library it exists. you can also just browse at random, but that doesn't really show you anything interesting (as you would expect given it's all random).
- tetris11 2y ago
  the hashing algorithm should encode some locality, but disappointingly doesn't... ...and can't because there is no original corpus that the locality hashing algorithm can use as a basis

sammex 2y ago

Would the index number actually be smaller than the actual data?

psychoslave 2y ago
You need both index and length, I guess. If concatenating both value is not enough to gain sufficient size shrink, you can always prefix a "number of times still needed to recursively de-index (repeat,start-point-index,size) concatenated triplets", and repeat until you match a desired size or lower. I don’t know if there would be any logical issue with this approach. The only logistical difficulty I can figure out is computing enough decimals and search the pattern in it, but I guess that such a voluminous pre-computed approximation can greatly help.
- waldrews 2y ago
  No invertible function can map every non-negative integer to a lower or equal non-negative integer (no perfect compression), but you can have functions that compress everything we care about at the cost of increasing the size of things we don't care about. So the recursive de-indexing strategy has to sometimes fail and increase the cost (once you account for storing the prefix).
  - psychoslave 2y ago
    Is there some inductive proof of that? Or is that some conjuncture? Actually any resources related to that point could be fun to explore
    - waldrews 2y ago
      It's a classic application of the pigeonhole principle, the first on in this list: https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_applications https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_...
waldrews 2y ago
It would average the same size as the actual data. Treating the pi bit sequence as random bits, and ignoring overlap effects, the probability that a given n bit sequence is the one you want is 1/2^n, so you need to try on average 2^n sequences to find the one you want, so the index to find it is typically of length n, up to some second order effects having to do with expectation of a log not being the log of an expectation.

Moosturm 2y ago

https://github.com/philipl/pifs https://github.com/philipl/pifs

maxmouchet 2y ago

https://news.ycombinator.com/item?id=8018818 https://news.ycombinator.com/item?id=8018818 and https://github.com/philipl/pifs https://github.com/philipl/pifs :-)

IsTom 2y ago

There are many ways in which a number might not never repeat itself, but not contain all sequences (e.g. never use a specific digit). What you want is normal numbers and pi is not proven to be one (though probably it is).

voytec 2y ago

> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. You reminded me of this Person of Interest clip: https://www.youtube.com/watch?v=fXTRcsxG7IQ https://www.youtube.com/watch?v=fXTRcsxG7IQ

A_D_E_P_T 2y ago

> every piece of conceivable information (music, movies, texts) is in there, encoded Borges wrote a famous short story, “The Library of Babel,” about a library where: “... each book contains four hundred ten pages; each page, forty lines; each line, approximately eighty black letters. There are also letters on the front cover of each book; these letters neither indicate nor prefigure what the pages inside will say. “There are twenty-five orthographic symbols. That discovery enabled mankind, three hundred years ago, to formulate a general theory of the Library and thereby satisfactorily resolve the riddle that no conjecture had been able to divine—the formless and chaotic nature of virtually all books. . . “Some five hundred years ago, the chief of one of the upper hexagons came across a book as jumbled as all the others, but containing almost two pages of homogeneous lines. He showed his find to a traveling decipherer, who told him the lines were written in Portuguese; others said it was Yiddish. Within the century experts had determined what the language actually was: a Samoyed-Lithuanian dialect of Guaraní, with inflections from classical Arabic. The content was also determined: the rudiments of combinatory analysis, illustrated with examples of endlessly repeating variations. These examples allowed a librarian of genius to discover the fundamental law of the Library. This philosopher observed that all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet. He also posited a fact which all travelers have since confirmed: In all the Library, there are no two identical books. From those incontrovertible premises, the librarian deduced that the Library is “total”—perfect, complete, and whole—and that its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite)—that is, all that is able to be expressed, in every language.” I've done the (simple) math on this -- in fact I'm writing a short book on the philosophy of mathematics where it's of passing importance -- and the library contains some 26^1312000 books, which makes 202T look like a very small number. So though everything you describe is encoded in Pi (assuming Pi is infinite and normal) we're a long, long way away from having useful things encoded therein... Also, an infinite and normal Pi absolutely repeats itself, and in fact repeats itself infinitely many times.

WillAdams 2y ago
And for an amusing example of this see: https://www.piday.org/find-birthday-in-pi/ https://www.piday.org/find-birthday-in-pi/

NooneAtAll3 2y ago

> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. may I interest you in the difference between *irrational* numbers and *normal* numbers? look at https://en.wikipedia.org/wiki/Liouville_number https://en.wikipedia.org/wiki/Liouville_number - no repeats, but minuscule "contained information"

euroderf 2y ago

> every piece of conceivable information (music, movies, texts) is in there, encoded. So that means that if we give a roomful of infinite monkeys an infinite number of hand-cranked calculators and an infinite amount of time, they will, as they calculate an infinite number of digits of pi, also reproduce the complete works of Shakespeare et al.

_joel 2y ago
and then do it all again, but backwards.

mkl 2y ago

> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. This is true for normal numbers [1], but is definitely not true for all non-repeating (irrational) numbers. Pi has not been proven to be normal. There are many non-repeating numbers that are not normal, for example 0.101001000100001... Storing the index into pi for a file would usually take something like as much space as just storing the file, and storing or calculating enough digits to use that index would be impossible with the technology of today (or even probably the next century). [1] https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number

tombert 2y ago
It's conjectured to be normal isn't it? I know it hasn't been proven yet, and I cannot seem to find where I read this, but I thought there was at least statistical evidence indicating that it's probably normal.
- adgjlsfhk1 2y ago
  100% of real numbers are normal, so that's pretty strong statistical evidence
  - hn_throwaway_99 2y ago
    What? No they're not, e.g. no rational numbers are normal, and they are real.
    - GraphEnthusiast 2y ago
      The rational numbers make up "zero percent" of the real numbers. It's a little hard to properly explain without assuming a degree in math, since the proper way to treat this requires measure theoretic probability (formally, the rationals have measure zero in the reals for the "standard" measure). The short version is that the size of the reals is a "bigger infinity" than the size of the rationals, so they effectively have 'zero weight'. Reference (very technical): https://math.stackexchange.com/questions/508217/showing-that-rationals-have-lebesgue-measure-zero#508225 https://math.stackexchange.com/questions/508217/showing-that...

sxv 2y ago

Isn't 202TB (for comparison) way too small to contain every permutation of information? That filesize wouldn't even be able to store a film enthusiast's collection?

RamblingCTO 2y ago
Well it all comes down to encoding, doesn't it. We can represent almost everything with just 0 and 1 as well, can't we? The description of that data is way bigger than the elements used to describe it of course.

_fizz_buzz_ 2y ago

This is not necessarily true. Pi might not repeat but it could at some point - for example - not contain the digit 3 anymore (or something like that). It would never repeat, but still not have all conceivable information.

pilaf 2y ago
But the number 3 is there just because we decide to calculate digits in base 10. We could encode Pi in binary instead, and since it doesn't repeat it necessarily will never be a point where there will never be another 1 or a 0, right?
- bubblyworld 2y ago
  That's true - you can quite easily prove that an eventually constant sequence of decimals codes for a rational number. But it's also true that pi may not contain every _possible_ sequence of decimals, no matter what base you pick. Like the Riemann hypothesis, it seems very likely and people have checked a lot of statistics, but nobody has proven it beyond a (mathematical) shadow of doubt.
- _fizz_buzz_ 2y ago
  Obviously, it was just an example to illustrate what a non-periodic number could look like that doesn’t contain all possible permutations. If the number never contains the digit 3 in base 10 it will also not contain all possible permutations in all other bases.
  - deleted 2y ago
    [deleted]

criddell 2y ago

> every piece of conceivable information is in there Wouldn't the encoded information have to have a finite length? For example, pi doesn't contain e, does it?

tzs 2y ago
> For example, pi doesn't contain e, does it? Assuming we are only interested in base 10 and that pi contains e means that at some point in the sequence of decimal digits of pi (3, 1, 4, 1, 5, 9, 2, ...) there is the sequence of decimal digits of e (2, 7, 1, 8, 2, 8, ...), then I believe that question is currently unanswered. Pi would contain e if and only if there are positive integers n and m such that 10^n pi - m = e, or equivalently 10^n pi - e = m. We generally don't know if combinations of e and pi of the form a pi + b e where a and b are algebraic are rational or not. Even the simple pi + e is beyond current mathematics. All we've got there is that at least one of pi + e and pi e must be irrational. We know that because both pi and e are zeros of the polynomial (x-pi)(x-e) = x^2 - (pi+e)x + pi e. If both pi+e and pi e were rational then that polynomial would have rational coefficients, and the roots of a non-zero polynomial with rational coefficients are algebraic (that is in fact the definition of an algebraic number) and both pi and e are known to not be algebraic.
RamblingCTO 2y ago
I implied that, yes

worewood 2y ago

The sad thing is that the index would take just as much space as the data itself, because in average you can expect to find a n-bit string at the 2^n position.

Zambyte 2y ago

https://news.ycombinator.com/item?id=36357466 https://news.ycombinator.com/item?id=36357466

2OEH8eoCRo0 2y ago

Does pi contain pi?

schoen 2y ago
It does, starting right at the beginning!

sundry_gecko 2y ago

Reminds me of a scene of Finch teaching in Person of Interest. https://m.youtube.com/watch?v=yGmYCfWyVAM https://m.youtube.com/watch?v=yGmYCfWyVAM

constantcrying 2y ago

>As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. It is somewhat shocking that again and again this logical fallacy comes up. Why do people think that this is true? It doesn't even sound true.

hkhanna 2y ago
Isn't it a property of infinity? If pi goes on infinitely without repeating itself, every possible combination of numbers appears somewhere in pi. It's sort of like the idea that if the universe is infinitely big and mass and energy are randomly distributed throughout the universe, then an exact copy of you on an exact copy of Earth is out there somewhere. This property of infinity has always fascinated me, so I'm very curious for where the logical fallacy might be.
- n2d4 2y ago
  Not necessarily. The number 1.01001000100001000001... never repeats itself, yet most other numbers can never be found in it. A number that contains all other numbers infinitely many times (uniformly) would be called normal, but no one has managed to prove this for pi yet. In fact, no one even managed to prove that pi doesn't contain only 0s and 1s like the above after the X-th digit.
- constantcrying 2y ago
  >Isn't it a property of infinity? If pi goes on infinitely without repeating itself, every possible combination of numbers appears somewhere in pi. No. Example: 0.1011011101111011111... does never repeat, yet there is no 2 in there, neither is there 00 in there.
  - onion2k 2y ago
    The fact you can't encode arbitrary data in a structured-but-irrational number doesn't mean you can't encode data in a 'random' irrational number. The question is really 'Does every series of numbers of arbitrary finite length appear in pi?' I can't answer that because I'm not a mathematician, but I also can't dismiss it, because I'm not a mathematician. It sounds like a fair question to me.

its_ethan 2y ago

https://libraryofbabel.info/ https://libraryofbabel.info/ you might find this to be pretty cool. It's similar to what you're describing. Whoever made it has an algorithm where you can look up "real" strings of text and it'll show you where in the library it exists. you can also just browse at random, but that doesn't really show you anything interesting (as you would expect given it's all random).

tetris11 2y ago
the hashing algorithm should encode some locality, but disappointingly doesn't... ...and can't because there is no original corpus that the locality hashing algorithm can use as a basis