5 ms·

Tupper's Self-Referential Formula

66 points by muon 16y ago

long 16y ago
this is awesome. how'd you stumble onto it?
- muon 16y ago
  I found this via Futility Closet, which a constant source of really interesting stuff. (http://www.futilitycloset.com/ http://www.futilitycloset.com/) Initially, I thought it's too good to be true, later found references in Wikipedia and Mathworld, and finally started to believe in it.
  - eru 16y ago
    Believe in it? This is math, you can check.
    - jtbigwoo 16y ago
      Normal people don't have the tools to check the math on a formula that involves 500-digit numbers. It's not like I'm going to pop out my TI-80 and graph it. This formula is cool, but it's only a bit more accessible than the human genome.
      - orangecat 16y ago
        Normal people don't have the tools to check the math on a formula that involves 500-digit numbers. Sure we do, for flexible values of "normal": from decimal import getcontext, Decimal getcontext().prec = 600 n = Decimal(960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719) # ignore floors, just using integer arguments f = lambda x,y: (y/17 * Decimal(2)**(-17*x - (y%17))) % 2 results = {} dhalf = Decimal('0.5') for x in xrange(106): for y in xrange(17): # this takes a while results[x,y] = '*' if f(Decimal(x), Decimal(y+n))>dhalf else ' ' # this prints reversed for some reason unless we reverse the x iteration for y in xrange(17): print(''.join(results[x,y]+' ' for x in xrange(105,-1,-1)))
        eru 16y ago
        Indeed. And lots of computers come with Python or something equally capable of arithmetic installed nowadays. (And if you don't have the means to check it, it's probably better to remain agnostic, instead of starting to believe random things.)
    - muon 16y ago
      Believe the blog post, I don't doubt the math.
      - eru 16y ago
        OK.
senki 16y ago
"The formula itself is a general purpose method of decoding a bitmap stored in the constant n, so it could actually be used to draw any other image, and does not in fact contain any reference to itself." http://en.wikipedia.org/wiki/Tuppers_self-referential_formula http://en.wikipedia.org/wiki/Tuppers_self-referential_formul...
- 10ren 16y ago
  As a high school student, I thrilled to the idea that long numbers (eg. pi digits) might contain messages. As a postgrad, I now know that the long numbers are actually a form of information (eg. a digital photograph is just a number: a sequence of bits). Sadly, there's nothing exciting about it, as the interesting sequences occur very sparsely, just as the vast majority of bit sequences (out of all those possible) represent uninteresting digital photographs.
  - senki 16y ago
    Yes, the PI really contains all information on the world, since it's an infinite non-periodic number. The only problem is that it also contains all noise on the world, too. It's like a block of marble, that contains all great sculptures, you just have to remove bits from here and there...
    - sp332 16y ago
      Pi doesn't contain all the information in the world. It doesn't have the digits of the square root of 2, or the value of e, for example.
      - wtallis 16y ago
        If you assume that pi is a normal number (unproven, but likely) then it is almost certain (ie. probability 1) that the digits of e and the square root of 2 are subsequences of the digits of pi.
        16y ago
aaront 16y ago
This is one of my SE prof's close colleagues. He showed this in class.
zackattack 16y ago
Someone wanna prove this on WolframAlpha.com?
mattsouth 16y ago
That is a big number. This reminds me of the old adage about monkeys and typewriters eventually writing Shakespeare's plays (http://en.wikipedia.org/wiki/Infinite_monkey_theorem http://en.wikipedia.org/wiki/Infinite_monkey_theorem)

66 points by muon 16y ago

long 16y ago

this is awesome. how'd you stumble onto it?

muon 16y ago
I found this via Futility Closet, which a constant source of really interesting stuff. (http://www.futilitycloset.com/ http://www.futilitycloset.com/) Initially, I thought it's too good to be true, later found references in Wikipedia and Mathworld, and finally started to believe in it.
- eru 16y ago
  Believe in it? This is math, you can check.
  - jtbigwoo 16y ago
    Normal people don't have the tools to check the math on a formula that involves 500-digit numbers. It's not like I'm going to pop out my TI-80 and graph it. This formula is cool, but it's only a bit more accessible than the human genome.
    - orangecat 16y ago
      Normal people don't have the tools to check the math on a formula that involves 500-digit numbers. Sure we do, for flexible values of "normal": from decimal import getcontext, Decimal getcontext().prec = 600 n = Decimal(960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719) # ignore floors, just using integer arguments f = lambda x,y: (y/17 * Decimal(2)**(-17*x - (y%17))) % 2 results = {} dhalf = Decimal('0.5') for x in xrange(106): for y in xrange(17): # this takes a while results[x,y] = '*' if f(Decimal(x), Decimal(y+n))>dhalf else ' ' # this prints reversed for some reason unless we reverse the x iteration for y in xrange(17): print(''.join(results[x,y]+' ' for x in xrange(105,-1,-1)))
      - eru 16y ago
        Indeed. And lots of computers come with Python or something equally capable of arithmetic installed nowadays. (And if you don't have the means to check it, it's probably better to remain agnostic, instead of starting to believe random things.)
  - muon 16y ago
    Believe the blog post, I don't doubt the math.
    - eru 16y ago
      OK.

senki 16y ago

"The formula itself is a general purpose method of decoding a bitmap stored in the constant n, so it could actually be used to draw any other image, and does not in fact contain any reference to itself." http://en.wikipedia.org/wiki/Tuppers_self-referential_formula http://en.wikipedia.org/wiki/Tuppers_self-referential_formul...

10ren 16y ago
As a high school student, I thrilled to the idea that long numbers (eg. pi digits) might contain messages. As a postgrad, I now know that the long numbers are actually a form of information (eg. a digital photograph is just a number: a sequence of bits). Sadly, there's nothing exciting about it, as the interesting sequences occur very sparsely, just as the vast majority of bit sequences (out of all those possible) represent uninteresting digital photographs.
- senki 16y ago
  Yes, the PI really contains all information on the world, since it's an infinite non-periodic number. The only problem is that it also contains all noise on the world, too. It's like a block of marble, that contains all great sculptures, you just have to remove bits from here and there...
  - sp332 16y ago
    Pi doesn't contain all the information in the world. It doesn't have the digits of the square root of 2, or the value of e, for example.
    - wtallis 16y ago
      If you assume that pi is a normal number (unproven, but likely) then it is almost certain (ie. probability 1) that the digits of e and the square root of 2 are subsequences of the digits of pi.
      - 16y ago

aaront 16y ago

This is one of my SE prof's close colleagues. He showed this in class.

zackattack 16y ago

Someone wanna prove this on WolframAlpha.com?

mattsouth 16y ago

That is a big number. This reminds me of the old adage about monkeys and typewriters eventually writing Shakespeare's plays (http://en.wikipedia.org/wiki/Infinite_monkey_theorem http://en.wikipedia.org/wiki/Infinite_monkey_theorem)