[math-fun] Messages in pi (was Re: math novels?)
Dan Asimov <dasimov@earthlink.net> wrote:
The thing I mainly recall about Contact is that a message is sent by somehow *altering* the digits of pi.
You are misremembering. There's no implication that pi had ever had a different value. (At least not in the paperback. Does anyone have a first edition hardback they can check?) Dave Dyer <ddyer@real-me.net> wrote:
On the other hand, since PI has an infinite number of nonrepeating digits, such finite patterns MUST exist.
That doesn't follow. Last I heard, nobody had ever proven pi normal. Normality, despite its name, is very weird. It's known that "nearly all" numbers are normal, i.e. non-normal numbers are of measure zero, but, to the best of my knowledge, nobody has ever proven any particular number to be normal. (Some numbers contrived for the purpose have been proven normal in one base, but not in all positive integer bases at once.) If pi *is* normal, as it probably is, then it's true that pi in binary must contain all possible bit strings, including copies of every book and DVD -- and the complete archives of this list, including posts that haven't been written yet. But of course all of these are buried enormously deep. Finding anything "interesting" in the first trillion digits would be, well, interesting. Also see http://www.netfunny.com/rhf/jokes/01/Jun/pi.html which is apparently the best known thing I've ever written. If pi is normal, does it necessarily contain ASCII text of a proof that pi is normal? :-) James Propp <jamespropp@gmail.com> wrote:
I wonder whether Sagan was falling prey to the confusion about whether pi is a mathematical constant or a physical constant.
Possibly. But he was obviously aware that it could be calculated, not just measured. A dimensioned physical constant such as the speed of light could be made to contain any desired message by a choice of how our system of units is set up. A dimensionless constant such as the fine structure constant or the proton-to-electron mass ratio would be the same number in any system of units. But those are of course measured, not calculated. They're both only known to about nine significant digits, and it would be an enormous amount of work to learn each additional digit, so there's no possibility of storing lengthy messages in them that we could read even if someone had write access to them. Anyhow, a message in a physical constant would merely prove that the writer is cleverer or more powerful than us, not that he is infinitely clever or powerful. A message in pi would prove the latter. Except that we could never rule out the possibility that our computers which "found" the message had been hacked. And that's a logically preferred hypothesis, since it would require only a finite amount of cleverness and power. Similarly for *any* alleged proof of a being's omniscience or omnipotence.
Personally, I think we should look for a message from God in the digits of seventeen. I mean, everyone says that after the decimal point there's just a whole lot of 0's --- but how far out have they looked?
No, the digits of seventeen *in base pi*, 120.220021101020230020003... This of course never repeats or terminates. And I'd bet that nobody has ever checked it for hidden messages. :-) As an aside, half the time this list seems to be more about Mathematica than about math. To how many digits can Mathematica calculate 17 in base pi? (I calculated the above digits using plain old C. It took me all of ten minutes to write the code.)
Keith F. Lynch: "No, the digits of seventeen *in base pi*, 120.220021101020230020003… This of course never repeats or terminates." MathWorld's article on 'base' cautions that "the representation of a given integer in an irrational base may be nonunique" (but this is true also of integer bases: 1=.999… in base ten). More relevant perhaps is the issue of normalcy. My understanding of the concept is that it cannot be applied to irrational bases. What then is the frequency distribution of the digits (0, 1, 2, 3) of seventeen in base pi?
If we're talking about fractional bases, allow me to introduce you to a conjecture of Leonard J. Schulman and me: Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be an integer, and let 0 = \sum_{i=0}^\infty a_i (2/3)^i . Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0 = 1. Thus the first term is 1, and the remaining terms need to cancel it out. Conjecture: all initial subsequences have a nonzero density of nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero. I am equally interested in any nearby conjecture: e.g. replace 3 with 5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on. If you can prove this, it will have nice consequences for a construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007 - Cris On Oct 13, 2013, at 12:24 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
Keith F. Lynch: "No, the digits of seventeen *in base pi*, 120.220021101020230020003… This of course never repeats or terminates."
MathWorld's article on 'base' cautions that "the representation of a given integer in an irrational base may be nonunique" (but this is true also of integer bases: 1=.999… in base ten). More relevant perhaps is the issue of normalcy. My understanding of the concept is that it cannot be applied to irrational bases. What then is the frequency distribution of the digits (0, 1, 2, 3) of seventeen in base pi? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0? Rich ----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0 = 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with 5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris
Sorry. I mean that there is a uniform constant C > 0 such that all representations of 0 with a_0=1 and a_i \in {-1,0,+1} have density at least C (in all initial subsequences). Cris On Oct 15, 2013, at 2:56 PM, rcs@xmission.com wrote:
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0?
Rich
----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0 = 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with 5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris
Sorry I'm late to this game, just trying to wrap my head around balanced base 3/2. Can you explicitly write down any nontrivial representation of 0? Should it be obvious to me what the "greedy" representation of 0 starting with a_0=1 is? Clearly I should read your paper, though this tastes enough like elliptic theta that I'm already a little scared. --Michael On Tue, Oct 15, 2013 at 5:31 PM, Cris Moore <moore@santafe.edu> wrote:
Sorry. I mean that there is a uniform constant C > 0 such that all representations of 0 with a_0=1 and a_i \in {-1,0,+1} have density at least C (in all initial subsequences).
Cris
On Oct 15, 2013, at 2:56 PM, rcs@xmission.com wrote:
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0?
Rich
----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to
a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be
an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0
= 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of nonzero
coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with
5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a
construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
I suppose the greedy representation is to set a_i = +1 or -1 depending on whether the sum we have so far is positive or negative. But we can get away with a lot of zeros if we achieve a partial sum that's quite close to zero... we don't need to use nonzero a_i until we get nervous about whether the terms are decreasing too fast in absolute value to get to zero from the current partial sum. Cris On Oct 19, 2013, at 8:54 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Sorry I'm late to this game, just trying to wrap my head around balanced base 3/2. Can you explicitly write down any nontrivial representation of 0? Should it be obvious to me what the "greedy" representation of 0 starting with a_0=1 is?
Clearly I should read your paper, though this tastes enough like elliptic theta that I'm already a little scared.
--Michael
On Tue, Oct 15, 2013 at 5:31 PM, Cris Moore <moore@santafe.edu> wrote:
Sorry. I mean that there is a uniform constant C > 0 such that all representations of 0 with a_0=1 and a_i \in {-1,0,+1} have density at least C (in all initial subsequences).
Cris
On Oct 15, 2013, at 2:56 PM, rcs@xmission.com wrote:
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0?
Rich
----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to
a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be
an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0
= 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of nonzero
coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with
5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a
construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
Oh, the "greedy representation" I was thinking about wasn't focused on zeros specifically -- rather it just appended whichever of {-1,0,+1} brought the partial sum closer to the target. (Presumably in case of a tie you would append a 0.) There's also the "greedily use 0's" representation, in which your next digit is a 0 unless that would render the goal unreachable. --Michael On Sat, Oct 19, 2013 at 11:55 AM, Cris Moore <moore@santafe.edu> wrote:
I suppose the greedy representation is to set a_i = +1 or -1 depending on whether the sum we have so far is positive or negative. But we can get away with a lot of zeros if we achieve a partial sum that's quite close to zero... we don't need to use nonzero a_i until we get nervous about whether the terms are decreasing too fast in absolute value to get to zero from the current partial sum.
Cris
On Oct 19, 2013, at 8:54 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Sorry I'm late to this game, just trying to wrap my head around balanced base 3/2. Can you explicitly write down any nontrivial representation of 0? Should it be obvious to me what the "greedy" representation of 0 starting with a_0=1 is?
Clearly I should read your paper, though this tastes enough like elliptic theta that I'm already a little scared.
--Michael
On Tue, Oct 15, 2013 at 5:31 PM, Cris Moore <moore@santafe.edu> wrote:
Sorry. I mean that there is a uniform constant C > 0 such that all representations of 0 with a_0=1 and a_i \in {-1,0,+1} have density at
least
C (in all initial subsequences).
Cris
On Oct 15, 2013, at 2:56 PM, rcs@xmission.com wrote:
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0?
Rich
----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to
a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i
be an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0
= 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of
nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with
5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a
construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute
The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
right. that achieves some constant fraction of zeros. the question is whether this constant can tend to 1. Cris On Oct 19, 2013, at 10:41 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
There's also the "greedily use 0's" representation, in which your next digit is a 0 unless that would render the goal unreachable.
--Michael
I suppose you'd just have to be more careful with the definition. The digits 0, 1, and 2 should occur with frequency 1/pi, while 3 should occur with frequency 1-3/pi. The strings 00, 01, 02, 10, 11, 12, 20, 21, 22 should occur with frequency 1/pi^2, etc. I imagine all the proofs about non-normal numbers having measure 0 would carry over mutatis mutandis. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Oct 13, 2013 at 2:24 PM, Hans Havermann <gladhobo@teksavvy.com>wrote:
Keith F. Lynch: "No, the digits of seventeen *in base pi*, 120.220021101020230020003… This of course never repeats or terminates."
MathWorld's article on 'base' cautions that "the representation of a given integer in an irrational base may be nonunique" (but this is true also of integer bases: 1=.999… in base ten). More relevant perhaps is the issue of normalcy. My understanding of the concept is that it cannot be applied to irrational bases. What then is the frequency distribution of the digits (0, 1, 2, 3) of seventeen in base pi? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (7)
-
Charles Greathouse -
Cris Moore -
Dan Asimov -
Hans Havermann -
Keith F. Lynch -
Michael Kleber -
rcs＠xmission.com