4 ms·

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a p

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005). [1] If pi is normal (as it is conjectured to be), then yeah it has to. [1] https://en.wikipedia.org/wiki/Normal_number#Properties https://en.wikipedia.org/wiki/Normal_number#Properties

criddell 7y ago
This is amazing! So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy! Do you know if k has to be finite?
- SamBam 7y ago
  K would have to be finite, yes, or there could never be another string after it. But k can be arbitrarily large. Yes, Normal Numbers are awesome to think about when you first learn them. Ever tiny detail of everything that has ever happened in the history of the world, written in incredibly beautiful prose that would bring the entire planet to its knees on reading it, is somewhere in the digits of pi. Of course, so are a lot of lies -- every possible variant of lie. And lots and lots and lots of junk. Indeed, you can make a program to search pi for any string of letters, and if your program is powerful enough you may find a 10 or 11 digit sequence of your choosing, but no more than that. And of course, searching for the string you pre-chose seems a lot more underwhelming than stumbling across the story of your life. https://www.angio.net/pi/ https://www.angio.net/pi/
  - criddell 7y ago
    That's really fascinating. So if you could find things in pi (or any normal number) quickly enough, then any arbitrary chunk of data could be compressed to just an offset and a length into that number. That's kind of crazy to think about.
    - dTal 7y ago
      No, not compressed - the offset would always be (much) larger than the arbitrary chunk of data. It's flawed intuition to think of the information as "in" pi - it's "in" the offset, and pi is essentially a small decryption algorithm. Don't get carried away with pi mysticism - all this stuff is also true of the concatenation of all integers in order (0123456789101112131415...).
    - andreareina 7y ago
      It's an idea as old as time. Or 8 years, at the very least: https://github.com/philipl/pifs https://github.com/philipl/pifs In expectation, a k-digit sequence will require a k-digit number to encode its offset, so there's no compression there. Then again, there's this gem 762 decimal places in: http://mathworld.wolfram.com/FeynmanPoint.html http://mathworld.wolfram.com/FeynmanPoint.html
      - SamBam 7y ago
        > https://github.com/philipl/pifs https://github.com/philipl/pifs That's brilliant. I also like that you don't need to store the offsets anywhere else, because that metadata is also just data that can itself be found in pi. Actually, that sounds like one of the sects in Library of Babel, who tried to find the one true catalogue (or whatever it was) by looking for the book that told where it was, and the book that told where that book was, etc. There must be an infinite number of books that lead the way to the catalogue.
jonathanstrange 7y ago
Conjectured, but still not proved to be normal. Or did I miss something this big?

criddell 7y ago

This is amazing! So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy! Do you know if k has to be finite?

SamBam 7y ago
K would have to be finite, yes, or there could never be another string after it. But k can be arbitrarily large. Yes, Normal Numbers are awesome to think about when you first learn them. Ever tiny detail of everything that has ever happened in the history of the world, written in incredibly beautiful prose that would bring the entire planet to its knees on reading it, is somewhere in the digits of pi. Of course, so are a lot of lies -- every possible variant of lie. And lots and lots and lots of junk. Indeed, you can make a program to search pi for any string of letters, and if your program is powerful enough you may find a 10 or 11 digit sequence of your choosing, but no more than that. And of course, searching for the string you pre-chose seems a lot more underwhelming than stumbling across the story of your life. https://www.angio.net/pi/ https://www.angio.net/pi/
- criddell 7y ago
  That's really fascinating. So if you could find things in pi (or any normal number) quickly enough, then any arbitrary chunk of data could be compressed to just an offset and a length into that number. That's kind of crazy to think about.
  - dTal 7y ago
    No, not compressed - the offset would always be (much) larger than the arbitrary chunk of data. It's flawed intuition to think of the information as "in" pi - it's "in" the offset, and pi is essentially a small decryption algorithm. Don't get carried away with pi mysticism - all this stuff is also true of the concatenation of all integers in order (0123456789101112131415...).
  - andreareina 7y ago
    It's an idea as old as time. Or 8 years, at the very least: https://github.com/philipl/pifs https://github.com/philipl/pifs In expectation, a k-digit sequence will require a k-digit number to encode its offset, so there's no compression there. Then again, there's this gem 762 decimal places in: http://mathworld.wolfram.com/FeynmanPoint.html http://mathworld.wolfram.com/FeynmanPoint.html
    - SamBam 7y ago
      > https://github.com/philipl/pifs https://github.com/philipl/pifs That's brilliant. I also like that you don't need to store the offsets anywhere else, because that metadata is also just data that can itself be found in pi. Actually, that sounds like one of the sects in Library of Babel, who tried to find the one true catalogue (or whatever it was) by looking for the book that told where it was, and the book that told where that book was, etc. There must be an infinite number of books that lead the way to the catalogue.

jonathanstrange 7y ago

Conjectured, but still not proved to be normal. Or did I miss something this big?