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    Thread: Infinite diagram G

    1. #1
      DEATH TO FANATICS! StonedApe's Avatar
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      Infinite diagram G

      I've been reading the book Godel Escher Bach: an eternal golden braid and while I can understand most of the concepts in the book some of the math is a little fuzzy.

      The author presents as an example an infinite tree diagram "diagram G". In two of the four nodes of the diagram he writes the letter G to stand for a copy of diagram G. In this way the diagram can be expanded indefinetly. I'll post an image if I can figure out how.

      The formula is:
      G(n)=n-G(G(n-1) for n>0
      G(0)=0

      The paragraph below this starts

      "How does this function G(n) code for the tree structure? Quite simply, if you construct a tree by placing G(n) below n, for all values n, you will recreate diagram G."

      What does he mean by placing G(n) below n? And what is the difference between G(n) and n, aren't they the same value?
      157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.

      Women and rhythm section first - Jaco Pastorious

    2. #2
      Xei
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      Could you post the page number, assuming you have the 20th anniversary edition?

    3. #3
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      N
      | - - - |
      n1- - n2
      - | - - |
      na,nb nc,nd

      Probably?
      I stomp on your ideas.

    4. #4
      Member ChaybaChayba's Avatar
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      I think hes talking about recursivity resulting into infinity. Look up recursivity and you will understand Diagram G. I haven't read that book tho, but I'm pretty sure this is what he is talking about.
      "Reject common sense to make the impossible possible." -Kamina

    5. #5
      Member ChaybaChayba's Avatar
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      Out of interest I downloaded the ebook and started reading so I thought I might as well post the diagrams for those who are interested:


      "Reject common sense to make the impossible possible." -Kamina

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      Member Wildman's Avatar
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      I may be wrong, but here's what I think. If you look at the bottom picture ChaybaChayba posted, you see that for each circular node (which corresponds to a given n), there is a line under it leading to another node -- that node is G(n) for that n. Thus, 1 is under 2 so G(2) is 1.

      G(n) is not necessarily equal to n -- for example, G(5) = 5 - G(G(4)) = 5 - G(3) = 5 - 2 = 3 and G(21) = 13.

      Hopefully I am right and this helps. Amazing book, by the way.
      PhilosopherStoned likes this.

    7. #7
      DEATH TO FANATICS! StonedApe's Avatar
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      It starts on page 135
      157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.

      Women and rhythm section first - Jaco Pastorious

    8. #8
      Rational Spiritualist DrunkenArse's Avatar
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      Wildman pretty much explained this. It's an example of a recurrence relation and is a way to define a sequence.

      This is a pretty cool sequence in that every integer can be written as G(n) for some n and there can only be two such integers n. This means that the tree will have no leafs (i.e. dead ends) and that there will never be more than two branches from a node.



      So The proof isn't complete. I'm gonna finish it tomorrow before I live up to my name. My assertions are absolutely correct though.
      Spoiler for Proof:




      Spoiler for First hundred values:


      Spoiler for Python code to calculate values:
      Last edited by PhilosopherStoned; 01-11-2011 at 01:44 PM.
      Xei likes this.

    9. #9
      I am become fish pear Abra's Avatar
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      Guys. Hey guys?

      Could we...

      Could we have a 'G.E.B. General' thread?
      Abraxas

      Quote Originally Posted by OldSparta
      I murdered someone, there was bloody everywhere. On the walls, on my hands. The air smelled metallic, like iron. My mouth... tasted metallic, like iron. The floor was metallic, probably iron

    10. #10
      Xei
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      Sure. I'm reading it at the moment (I tried and failed a couple of years ago... wasn't intellectually ready) and it's brilliant.

    11. #11
      Rational Spiritualist DrunkenArse's Avatar
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      Post

      Ok, finally found time to do this. It made me really wish the forum had LaTeX but no such luck. If you actually want to read it, my condolences. It's a little longish but it's broken up into pieces and isn't too difficult. I would think that somebody with the mathematical maturity to handle calculus should be able to handle this (possibly with some questions). There's no calculus or algebra involved, this is completely elementary.

      Spoiler for Proof.:
      Previously PhilosopherStoned

    12. #12
      Xei
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      Yeah, I've read past this page now. It's really quite simple compared to some of the preceding stuff. Basically, if you pick a node, and work out G for that node, that value will be below that node.

    13. #13
      DEATH TO FANATICS! StonedApe's Avatar
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      I understand this now, I think I'm just really out of practice with math, does anyone know of any websites where I can get some algebra problems with solutions? I feel like I've forgotten everything from high school math.

      philosopherstoned, that link in your sig is hillarious.
      Last edited by StonedApe; 01-13-2011 at 06:25 PM.
      157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.

      Women and rhythm section first - Jaco Pastorious

    14. #14
      Xei
      UnitedKingdom Xei is offline
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      I'm sure a quick Google will get you a good site.

      I don't think you need to know much or any algebra to comprehend the book though.

      Did you mess around with the hyphen systems he made, though? That's more important I think, but it doesn't require maths as such, just basic reasoning.

    15. #15
      DEATH TO FANATICS! StonedApe's Avatar
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      I did, I understood how the diagram functioned right away, I just got confused when I started messing with the equation. I understand the book fine, I just used to be good with math and equations and am not at all anymore.
      157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.

      Women and rhythm section first - Jaco Pastorious

    16. #16
      Rational Spiritualist DrunkenArse's Avatar
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      For what it's worth, there's not a lot of algebraic manipulation to be done with the recurrence relation. The fact that these things are recursive means that when you try to do algebra with them, you first have to expand out one side. But then... surprise! You have to expand it out again because it's defined in terms of itself. ad infinitum.

      You'll note that when I was playing with it, I didn't solve for much of anything. For example, to conclude that

      G(n) - G(n - 2) = [G(n) - G(n - 1)] + [G(n - 1) + G(n - 2)]

      I just wrote

      G(n) - G(n - 2) = G(n) - G(n - 2)

      and added zero to the right hand side. Just remember that with addition, one can distribute parenthesis however one wants so long as sign changes are taken into account. So the right part could be written

      G(n) - [G(n - 1) - G(n - 1)] - G(n - 2)

      But

      G(n - 1) - G(n -1)

      is pretty plainly zero! One learns tricks like that by writing and reading proofs, not by studying basic high school "mathematics".
      Last edited by PhilosopherStoned; 01-13-2011 at 09:04 PM.
      Previously PhilosopherStoned

    17. #17
      Xei
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      Ad baseum, more like.

      Well, Hofstadter's pretty keen to stress that at this point, anyway.

    18. #18
      DuB
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      Quote Originally Posted by stonedape View Post
      does anyone know of any websites where I can get some algebra problems with solutions? I feel like I've forgotten everything from high school math.
      Try this:

      Khan Academy

    19. #19
      Rational Spiritualist DrunkenArse's Avatar
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      Yeah, ad baseum is definitely right. I'm not much on latin. It honestly might as well be ad infinitum for the simple reason that it's a huge pain in the ass to deal with one of these things fully expanded. ugghhh.

      EDIT:

      Of course the distinction between the two is that in a proof, one could possibly use the fact that it can be expanded to G(0) without actually doing so.
      Last edited by PhilosopherStoned; 01-13-2011 at 09:45 PM.
      Previously PhilosopherStoned

    20. #20
      Xei
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      Actually as far as I'm aware ad baseum is something I made up, ha. I'm sure there's a proper term for it but... yeah, dead language and all.

      The Khan Academy is good, good call.

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