Repeated digits in pi
Walter Nissen
Since my last post, I have learned a bit about this problem, but very
little. I thank each of the respondents for their help. This is what I
know. 3 is the first single digit in pi. 33 is the first doubled digit.
111 is the first tripled digit. From searches at Jeremy Gilbert's Web
page, http://gryphon.ccs.brandeis.edu/~grath/attractions/gpi/index.html, I
derive this table:
digits digit #
3 1
33 25
111 154
999999 763
3333333 710101
http://cad.ucla.edu:8001/amiinpi confirms the first part of this.
The remarkable repetition at digit #763 is called the Feynman point.
Perhaps because the late, great Richard P. Feynman called attention to
it??
I would welcome any information about extension of this table, especially
resources on the Net. What I have so far seems pitiful compared to the 4G
computed digits.
Thanks.
Cheers.
Walter Nissen dk...@cleveland.freenet.edu
P.S. Congratulations and thanks to Wei-Hwa Huang (whu...@cco.caltech.edu)
who responded to my:
> Concupiscence contains 4 'c's. Can you give a succinct word with 3 'c's?
with the sharp and delightful,
"No, but I found an unusual word with 3 'u's".
"coccyx" has to be the favorite positive response.
---
The 'c's in oscilloscope are pronounced differently. Can you give another
English word with 'sc's which are pronounced differently?
dave berner
> If I recall his auto-more-or-less-biography correctly, Feynman had the
> ambition of memorizing pi up to the six nines. That way, he could
> rattle off all 762 digits up to that point, and conclude "...nine,
> nine, nine, nine, nine, nine... and so on!"
>
> He didn't succeed, though.
actually i think this ancedote comes from Douglas Hofstadter.
I've been reading his book _Metamagical Themas_, and I think that this
passage is in the chapter about numeracy. (ch 4??? i don't have it
with me.)
-dave berner
Joseph Bork
Walter Nissen <dk...@cleveland.Freenet.Edu> wrote:
>
>Since my last post, I have learned a bit about this problem, but very
>little. I thank each of the respondents for their help. This is what I
>know. 3 is the first single digit in pi. 33 is the first doubled digit.
>111 is the first tripled digit. From searches at Jeremy Gilbert's Web
>page, http://gryphon.ccs.brandeis.edu/~grath/attractions/gpi/index.html, I
>derive this table:
>
> digits digit #
> 3 1
> 33 25
> 111 154
> 999999 763
>3333333 710101
>
I find this very similar to the conjecture that pi is a "normal" number. That
is, given the digits of pi, taken one at a time, each digit occurs 10% of the
time; taken two digits at a time, each double-digit combination ("11", "22",
"33", etc.) occurs 1% of the time; taken three digits at a time, triple-digit
combinations occur 0.1% of the time, and so forth. I do not have a sampling
of the digits of pi which is sufficiently large enough to demonstrate the
normality of pi for more than the single-digit case. (The small program I
wrote showed a distribution of 10% +/- 0.0001%) I believe the conjecture is
widely accepted, but as of yet unproven. If anyone is able to clarify the
definition of "normal," I would appreciate knowing exactly how to group the
digits of pi by two. For instance, given the first few digits of pi:
3.141592653589793
would the first group of two be "31" and then "41" or would it be "31" and
then "14"? At any rate, I just checked in the book _Islands of Truth: A
Mathematical Mystery Cruise_ by Ivars Peterson, and he has this interesting
tidbit to add:
"Checking the distribution of the 2 million poker hands in the first 10
million decimal digits of pi shows there is no significant deviation from
the expected values."
From the same book, table 5.19, page 183:
Poker Hand Expected number Actual number
--------------------------------------------------------------------------
no numbers the same 604,800 604,976
one pair 1,008,000 1,007,151
two pair 216,000 216,520
three of a kind 144,000 144,375
full house 18,000 17,891
four of a kind 9,000 8,887
five of a kind 200 200
That said, would you consider the occurance of the digits "999999" to be only
a 6-digit combination, but also a 1-, 2-, 3-digit combination, etc.?
In case you were wondering, pi is one of my favorite numbers. :>
>---
>The 'c's in oscilloscope are pronounced differently. Can you give another
>English word with 'sc's which are pronounced differently?
Hmm... "conscience"?
--Joe Bork
(jb...@apk.net)
Andrew C. Plotkin
> digits digit #
> 3 1
> 33 25
> 111 154
> 999999 763
> 3333333 710101
>
> The remarkable repetition at digit #763 is called the Feynman point.
> Perhaps because the late, great Richard P. Feynman called attention to
> it??
If I recall his auto-more-or-less-biography correctly, Feynman had the
ambition of memorizing pi up to the six nines. That way, he could
rattle off all 762 digits up to that point, and conclude "...nine,
nine, nine, nine, nine, nine... and so on!"
He didn't succeed, though.
--Z
"And Aholibamah bare Jeush, and Jaalam, and Korah: these were the borogoves..."
Andrew C. Plotkin
> In article <wkl8_IG00...@andrew.cmu.edu>, "Andrew C. Plotkin"
>
> > ambition of memorizing pi up to the six nines. That way, he could
> > rattle off all 762 digits up to that point, and conclude "...nine,
> > nine, nine, nine, nine, nine... and so on!"
> >
> > He didn't succeed, though.
>
> I've been reading his book _Metamagical Themas_, and I think that this
> passage is in the chapter about numeracy. (ch 4??? i don't have it
> with me.)
I reread both _Metamagical Themas_ and both Feynman books (_Surely
You're Joking..._ and _What Do You Care..._) recently, so it's very
possible I mixed up the reference. Sigh.
Bill Taylor
|> > digits digit #
|> > 3 1
|> > 33 25
|> > 111 154
|> > 999999 763
|>
|> I find this very similar to the conjecture that pi is a "normal" number. That
|> is, given the digits of pi, taken one at a time, each digit occurs 10% of the
|> time; taken two digits at a time, each double-digit combination ("11", "22",
|> "33", etc.) occurs 1% of the time; taken three digits at a time, triple-digit
|> combinations occur 0.1% of the time, and so forth.
Right. So it will not have gone unobserved that that SIX-digit repeat at
place 763 has come waaaaaaaaaay to early. Time for my favorite nonsense repost.
-------
Many folk have previously raised the point that any simple multiple of pi, might
equally well be considered the base constant, and pi to be just a multiple
of this. This idea struck a chord with others, and though it's too soon to tell,
I get the feeling that 2.pi might be the most popular choice (especially
if physicists get a vote too).
Certainly, most of us will have been struck by the fact that both 2.pi and
pi/2 appear far more often in math formulas than pi itself, and thus either
of these might have a better claim to be the 'basic' constant.
Let's see what God says.....
(Carl Sagan in 'Cosmos' suggested that we would eventually find a big circle
way down in the zillionsths places of pi's base-11 expansion; how fanciful...!)
I was struck by the following, in good old base 10, in the mere 700s places.
pi = 3.14159..... ..1134 999999 837....
^ ^
762nd dec place 767th dec place
Now in the first 1000 or so places, we could expect to find about ten cases
of some digit repeated 3 times; and some digit repeated 4 times, maybe once.
But here we have a digit repeated SIXFOLD, and that digit is a 9, of all things
(though a zero would do as well). ~~~~~~~
This means, if we do calculations with pi to 761 dec places (rounded to ..1135),
we get AN EXTRA SIX PLACES THROWN IN 'FREE' .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Surely this can't be a fluke !!!! So I think God is telling us that...
(i) we are right to be using the decimal system (so much for computer scientists
and British-unit cranks)
(ii) we should be calculating to 761 place accuracy.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now, let's go a little further..... Calculating from the above, we get
pi/4 = .......83 749999 959.....
pi/2 = ......567 499999 918.....
2 pi = .....2269 999999 67......
4 pi = .......39 999999 3.......
8 pi = .......79 999998 ........
Note that doubling and halving beyond these
just steadily makes matters worse. So we see that with 2.pi (or 4.pi) , we get
even one more decimal place for free !
So it seems clear then, that God's further instruction is...
(iii) we should be using 2.pi as our basic constant!
Please do not bother to email me about your personal messages from Hermit
on this matter...........
-------------
Please note the new PC pronoun "Hermit" for Him/Her/It.
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Luke: I just felt a major disturbance in the farce.
-------------------------------------------------------------------------------
MATTHEW PRIESTLEY
What you need is a program to calculate pi not based upon the machine's
memory capacities. The problem with pi programs is that they require
floating point accuracy to an incredible degree. So what I've done instead
is write a program that will carry out long division on *strings*, which
permits one digit of accuracy per byte of free memory. here's a sample:
3.141592653589793238462643383279502884197169399....
^^ ^^ ^^
neet o!
--
__ ___
/_/\ / /\
|_|:\ *************************************** / /::\
/ /::\ * MATTHEW PRIESTLEY * / /:/\:\
_/ /:/\:\ * prie...@uiuc.edu * / /:/ /:/
/_/:./::\ \:\ * http://www.cen.uiuc.edu/~priestle * /__/:/ /:/
\ \::/\:\ \:\ * * \ \:\/:/
\ \:\_\/__\/ * * \ \::/
\ \:\ * Rage is Optional. * \ \:\
\ \:\ * * \ \:\
\ \:\ *************************************** \ \:\
\__\/ \__\/
Brian and Margo Gordon
The authors took the first 200,000 combinations through the digits, and counted
them as poker hands...
31415 (a pair of ones)
14159 (a pair of ones)
41592 (garbage)
15926 (garbage)
59265 (a pair of fives)
etc...
The statistics for the different hands (one pair, two pair, three of a kind,
four of a kind, full house, straight, five of a kind -- sorry, no flushes!)
were almost perfectly as expected.
--bri
--
Blah blah blah internet blah blah purple blah blah finger-sandwich.
Team Skrub
Ilian P
> How did I stumble upon this