Happy Pi Day!

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  • Very compelling arguments there Phil, but, but... you are giving me headache...

    Let me try the following argument:

    1) Let's work in binary. Because we are supposed to be computer nerds and it's as good as any base.

    2) I suggest that as one wonders down the binary digits of PI one might find, somewhere along the way, that the digits start to look exactly like Tau. In fact they might continue to look like the digits of Tau forever and actually be the infinite digits of Tau. Does that thought break any rules? We still have random looking, "normal", gunk going on forever. We have not suddenly broken out into a rational number.

    3) But as you point out, in binary the digits of Tau contain the digits of PI, by a simple shift.

    Ergo, Pi can contain Tau, and you say Tau contains Pi. Which will again contain Pi, ad infinitum.

    There seems to be some paradox going on here...



  • heater wrote:
    Does that thought break any rules?
    No, but it's not guaranteed by pi's presumed normalcy. The only guarantee applies to finite strings of digits.

    In fact, pi contains an infinite number of infinite strings. You just pick which digit to start from. But, according to the rules of normalcy, the probability of finding a particular infinite string, picked at random, is zero, since it occurs with zero frequency.

    -Phil
    “Perfection is achieved not when there is nothing more to add, but when there is nothing left to take away. -Antoine de Saint-Exupery
  • Arghh... more headache!

    Thanks to Cantor we have have the idea that there are infinitely more real numbers along the number line than there is integers. If you were selecting numbers off the number line at random the probability of hitting an integer would be zero. But, we know already that there are integers in there!

    Similarly, I don't yet buy the idea that "probability of finding a particular infinite string, picked at random, is zero, since it occurs with zero frequency".

    The probability of finding it may be zero, but like the integers among the infinity of reals, it is still in there.

  • The probability of finding it may be zero, but like the integers among the infinity of reals, it is still in there.
    Maybe. But it's not a requirement of normalcy.

    -Phil
    “Perfection is achieved not when there is nothing more to add, but when there is nothing left to take away. -Antoine de Saint-Exupery
  • Here's another thought: does the decimal expansion of pi contain itself? No, it can't, because then it would be a repeating decimal, IOW rational. But pi is irrational.

    -Phil
    “Perfection is achieved not when there is nothing more to add, but when there is nothing left to take away. -Antoine de Saint-Exupery
  • Here's another rebuttal: the number of infinite subsequences in any expansion of pi (or any other number) is countable, since the number of digits in that expansion is countable, and you have to start from one of those digits. The number of real numbers, all of which can be expressed as infinite expansions, is uncountable. Therefore, there are an uncountably infinite number of infinite sequences that cannot be found as a subsequence of any particular infinite expansion, normal or otherwise.

    -Phil
    “Perfection is achieved not when there is nothing more to add, but when there is nothing left to take away. -Antoine de Saint-Exupery
  • I'm getting confused as to which "grade" of infinity you are using when, there.

    But you seem to be suggesting that there is an uncountably infinite number of countably infinitely long sequences than cannot be found in the digits of PI.

    PI itself being one. (Apart from the sequence starting at the beginning of course)

    Sounds reasonable.

    I'm going to take another beer and think about it...




  • ercoerco Posts: 19,515
    edited 2018-03-19 - 02:51:42
    I can't imagine PhiPi switching without a fight. Is there an irrational fellow anywhere who can go tau to tau with PhiPi on his best day?
    "When you make a thing, a thing that is new, it is so complicated making it that it is bound to be ugly. But those that make it after you, they don’t have to worry about making it. And they can make it pretty, and so everybody can like it when others make it after you."

    - Pablo Picasso
  • erco,
    I can't imagine PhiPi switching without a fight. Is there an irrational fellow anywhere who can go tau to tau with PhiPi on his best day?
    Perhaps we can get all the Phillip Tau's in the world to argue the case. For example: https://hypebloomington.wordpress.com/2015/02/02/featured-yp-of-the-month-phillip-tau/

    :)
  • it's been about a year since I got an external hard drive for my Raspberry PI. Purchased from Western Digital... 314 GB, introduced on March 14th for the price of $31.42. :smile:

    Founder of the "Society for Aimless Tinkering and World Conquest"
  • edited 2018-03-20 - 01:28:48
    The Argentinian author Jorge Luis Borges wrote a short story on this theme, "The Library of Babel," which is included in his anthology, Labyrinths. The library is of indefinite size, and its books contain every possible sequence of 25 lexical symbols. Borges riffs on this theme in exquisite detail, making it a very fun read.

    -Phil

    I always think of a 640 by 480 VGA screen with 256 colors. You could easily create a program to go through every sequence in order from all black to all white. You would be able to view every image ever produced and anything that ever could be produced. Not sure but I think there would be 256^(640*480) combinations. Might take a while, especially if you had to press the space bar to step through the series!


    Sandy

    Infantryman's Axiom; Always cheat, always win.
  • ercoerco Posts: 19,515
    edited 2018-03-20 - 18:24:29
    Heater. wrote: »
    Perhaps we can get all the Phillip Tau's in the world to argue the case.

    Touché! We just gotta get these kids together.

    Have to get back to you later, I'm watching a movie now.

    pi.jpg
    342 x 405 - 31K
    "When you make a thing, a thing that is new, it is so complicated making it that it is bound to be ugly. But those that make it after you, they don’t have to worry about making it. And they can make it pretty, and so everybody can like it when others make it after you."

    - Pablo Picasso
  • PublisonPublison Posts: 10,962
    edited 2019-03-14 - 13:37:17
    Infernal Machine
  • ercoerco Posts: 19,515
    Jim, you beat me to it! As far as I'm concerned, Boston Market and other restaurants have forever settled the whole "which is better, Pi or Tau" debate. See also https://www.thrillist.com/news/nation/pi-day-2019-deals-food-cheap-pizza

    This says 7-11 pizzas cost 3.14 today, although the in-store ad I saw said 2 for $7. Still cheaper better than gnawing on some nasty Tau all day IMHO.
    "When you make a thing, a thing that is new, it is so complicated making it that it is bound to be ugly. But those that make it after you, they don’t have to worry about making it. And they can make it pretty, and so everybody can like it when others make it after you."

    - Pablo Picasso
  • ercoerco Posts: 19,515
    Got 22 trillion decimal places? It was calculated using an Arduino.







    JK...

    https://www.pi2e.ch/

    Says NASA only uses 15 decimal places for rockets, and you'd only need 40 DPs for an atom-precise measurement of the universe. Now I almost feel silly for memorizing 50 DPs back in Jr. High, "except that chicks dig Pi", said no one ever.
    "When you make a thing, a thing that is new, it is so complicated making it that it is bound to be ugly. But those that make it after you, they don’t have to worry about making it. And they can make it pretty, and so everybody can like it when others make it after you."

    - Pablo Picasso
  • My access PIN is the last 4 digits of Pi, can you tell me what they are?
    Along with 'Antimatter' and 'Dark Matter' we've recently discovered the existence of
    `Doesn't Matter`, which appears to have no effect on the universe whatsoever.
  • ercoerco Posts: 19,515
    My access PIN is the last 4 digits of Pi, can you tell me what they are?

    6666.

    The Devil's in the details.
    "When you make a thing, a thing that is new, it is so complicated making it that it is bound to be ugly. But those that make it after you, they don’t have to worry about making it. And they can make it pretty, and so everybody can like it when others make it after you."

    - Pablo Picasso
  • erco wrote: »
    My access PIN is the last 4 digits of Pi, can you tell me what they are?

    6666.

    The Devil's in the details.

    Are you sure it isn't 9999, for those folks down-under...
    Along with 'Antimatter' and 'Dark Matter' we've recently discovered the existence of
    `Doesn't Matter`, which appears to have no effect on the universe whatsoever.
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