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.
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.
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.
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.
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.
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...
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.
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!
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.
Jim beat me to it! Happy Pi Day and everyone please stay safe. If you don't go out for Pie then stay in and spend your day running the Pi algorithm of your choice on the Parallax processor of your choice. The devil's in the details!
Jim beat me to it! Happy Pi Day and everyone please stay safe. If you don't go out for Pie then stay in and spend your day running the Pi algorithm of your choice on the Parallax processor of your choice. The devil's in the details!
I'm late to Netflix's "Stranger Things" party but daughter Amy loves it. It's set in the 1980's and suddenly she's interested in lots of my retro things, cassette tapes, Heathkit ham radios, my Nike leather Cortez, etc. Lots of nerdy references too. Planck's constant literally saves the day in the final episode of season 3. Turns out, the value is the combination to a safe which...
Comments
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...
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
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.
-Phil
-Phil
-Phil
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...
@
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
Touché! We just gotta get these kids together.
Have to get back to you later, I'm watching a movie now.
Local Boston Markets have a BOGO Pot pi(e) promotion going.
https://www.bostonmarket.com/promotions/3-14-19-pi-day-bogo-eblast/?utm_source=sfmc&utm_medium=email&utm_content=piday2019
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.
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.
6666.
The Devil's in the details.
Are you sure it isn't 9999, for those folks down-under...
Another year has come and gone.
https://en.wikipedia.org/wiki/Category:Pi_algorithms
Wondering if you were awake.
Don't make me say it...
We're celebrating Pi Day tomorrow with an extra hour of daylight! That's over 19*Pi minutes.
Sorta. Kinda. Ish. Though Tau lovers will probably say we're losing an hour...
I think the record for Pi is 31 trillion digits, depends on how much you want to split a hair. Goes on indefinitely.
Hapπ π-day 2021!
Stand-up Maths — Calculating π with Avogadro's Number
Ok... mayhaps the molecules aren't really square... but the result still is impressive for such slopπ assumptions!
My late wife (Wendy) made this pie for me back in March of 2012 in celebration of Pi Day.
I made this pizza pie...same night.
Wendy's counter-point to the phrase, "Pi R Round" was, "Cobbler R Square"
NICE Pie! And speaking of irrational constants...
I'm late to Netflix's "Stranger Things" party but daughter Amy loves it. It's set in the 1980's and suddenly she's interested in lots of my retro things, cassette tapes, Heathkit ham radios, my Nike leather Cortez, etc. Lots of nerdy references too. Planck's constant literally saves the day in the final episode of season 3. Turns out, the value is the combination to a safe which...
Yay, Pi day. we've got a nice chicken pot pie for dinner and a strawberry rhubarb pie for dessert.
https://www.usatoday.com/story/money/restaurants/2021/03/14/pi-day-2021-pizza-deals-time-change-daylight-saving/4680903001/