Re: [math-fun] Possibility puzzle
-------- Original Message -------- Subject: Re: [math-fun] Possibility puzzle Date: 2014-02-16 10:43 From: "Adam P. Goucher" <apgoucher@gmx.com> To: "math-fun" <math-fun@mailman.xmission.com>
Bill Gosper wrote:
Apparently, possibility is the intersection of probability with Boolean algebra. A certain 4th grade teacher decided to enrich Saxon Math with a few days of
What's Saxon Math? Does it involve the Angle bisector theorem?
reduced to bluffing with ambigrams: gosper.org/logic.png
I can remember an anecdote that in some Cambridge lecture on propositional logic, the lecturer wrote a few axioms such as this: (S ^ H) v N <==> (S v N) ^ (H v N) And then rotated the blackboard through 180° to give the remaining axioms. By the way, has anyone else realised that Soddy's hexlet generalises by adding extra layers of spheres? http://cp4space.wordpress.com/2014/02/15/soddys-hexlet/ Sincerely, Adam P. Goucher -------------- Frightful generalizations follow from http://mathworld.wolfram.com/BowlofIntegers.html . Related: http://www.tweedledum.com/rwg/Sodddy.htm The homeschoolers I know refer to Saxon Math as Drill and Kill, which has apparently switched from swords to pikes. --rwg
Funsters, Perhaps not surprising to mathematicians, functional MRI shows that beautiful equations trigger the same physiological response in mathematician's brains as beautiful art or music: http://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-ac... George http://georgehart.com/ P.S. My latest Mathematical Impressions video is on the beautiful math/music of change ringing: https://www.simonsfoundation.org/multimedia/mathematical-impressions-change-...
I suspect the same would apply to elegant programming - e.g. when someone first deduces or is introduced to a binary search algorithm..... On 19 Feb 2014, at 16:44, George Hart wrote:
Funsters,
Perhaps not surprising to mathematicians, functional MRI shows that beautiful equations trigger the same physiological response in mathematician's brains as beautiful art or music:
http://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-ac...
George http://georgehart.com/
P.S. My latest Mathematical Impressions video is on the beautiful math/music of change ringing: https://www.simonsfoundation.org/multimedia/mathematical-impressions-change-...
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
People tended to rate formulae they understood more highly. I think that I would have rated the BBP formula for pi quite low until someone pointed out that it allows calculating any hex digit of pi without the previous ones. Then I'd be fascinated. Is there a geometric way to see why the formula works or why the constants have the values they do? On Wed, Feb 19, 2014 at 10:09 AM, David Makin <makinmagic@tiscali.co.uk> wrote:
I suspect the same would apply to elegant programming - e.g. when someone first deduces or is introduced to a binary search algorithm.....
On 19 Feb 2014, at 16:44, George Hart wrote:
Funsters,
Perhaps not surprising to mathematicians, functional MRI shows that beautiful equations trigger the same physiological response in mathematician's brains as beautiful art or music:
http://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-ac...
George http://georgehart.com/
P.S. My latest Mathematical Impressions video is on the beautiful math/music of change ringing: https://www.simonsfoundation.org/multimedia/mathematical-impressions-change-...
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On Wed, Feb 19, 2014 at 11:34 AM, Mike Stay <metaweta@gmail.com> wrote:
People tended to rate formulae they understood more highly. I think that I would have rated the BBP formula for pi quite low until someone pointed out that it allows calculating any hex digit of pi without the previous ones. Then I'd be fascinated.
Is there a geometric way to see why the formula works or why the constants have the values they do?
Are you referring to this formula: https://24.media.tumblr.com/727f15001c4bc49b54bbb6abb1d9c868/tumblr_n19mynFg... Personally I find it quite nice, even without knowing that with it you can compute any hex digit of pi without the previous ones. But I did not see it on the list of 60 formulas presented in the article. I agree, that knowing more about a formula enhances ones feeling about it.
On Wed, Feb 19, 2014 at 10:09 AM, David Makin <makinmagic@tiscali.co.uk> wrote:
I suspect the same would apply to elegant programming - e.g. when someone first deduces or is introduced to a binary search algorithm.....
On 19 Feb 2014, at 16:44, George Hart wrote:
Funsters,
Perhaps not surprising to mathematicians, functional MRI shows that beautiful equations trigger the same physiological response in mathematician's brains as beautiful art or music:
http://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-ac...
George http://georgehart.com/
P.S. My latest Mathematical Impressions video is on the beautiful
math/music of change ringing:
https://www.simonsfoundation.org/multimedia/mathematical-impressions-change-...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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On Wed, Feb 19, 2014 at 4:17 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
On Wed, Feb 19, 2014 at 11:34 AM, Mike Stay <metaweta@gmail.com> wrote:
People tended to rate formulae they understood more highly. I think that I would have rated the BBP formula for pi quite low until someone pointed out that it allows calculating any hex digit of pi without the previous ones. Then I'd be fascinated.
Is there a geometric way to see why the formula works or why the constants have the values they do?
Are you referring to this formula:
https://24.media.tumblr.com/727f15001c4bc49b54bbb6abb1d9c868/tumblr_n19mynFg...
Yes, that's the one.
Personally I find it quite nice, even without knowing that with it you can compute any hex digit of pi without the previous ones. But I did not see it on the list of 60 formulas presented in the article. I agree, that knowing more about a formula enhances ones feeling about it.
Compared to, say, the Wallis or Gregory formulas, or the non-simple continued fraction for 4/pi (eqn 3 at http://mathworld.wolfram.com/PiContinuedFraction.html), it looked much less elegant to me at first glance. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Has anyone seen the expansion of π to inverse-factorial base? More precisely, I’d like to know the first N coefficients c_n of oo π-3 = Sum c_k / k! k=1 where 0 <= c_k < k, for N not too small. Even better, does anyone know a nice algorithm for computing this inverse-factorial base expansion of a real number that avoids over/underflow (without using arbitrary-precision arithmetic)? —Dan
On Wed, Feb 19, 2014 at 7:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Has anyone seen the expansion of π to inverse-factorial base?
More precisely, I'd like to know the first N coefficients c_n of
oo π-3 = Sum c_k / k! k=1
where 0 <= c_k < k, for N not too small.
Even better, does anyone know a nice algorithm for computing this inverse-factorial base expansion of a real number that avoids over/underflow (without using arbitrary-precision arithmetic)?
--Dan
It is easy to compute, with arbitrary precision arithmetic, but I don't know how without that. See OEIS: http://oeis.org/A007514
Hello, no, I don't think it does work, pi in factorial base, that is the whole point. I tried that for years, getting the n'th digit of e or Pi in factorial base, I was convinced that I could do the n'th decimal digit of e. There is a paper of Sierpinski about factorial bases, somewhere in his collected work, see vol 1. Many of the ideas in that paper where used later to make a general base algorithm. I have one with over 500 variants of that. The thing with these bases and representations is one of the reasons why I joined Neil Sloane in the adventure of the EIS, OEIS, and HIS to get the answer about that. Well, there are no definite answer. In the factorial base , the number E is 1, of course but Pi is messy and completely random. If you use Pi and one of the Newton, Gregory or any natural bases for that number , that's ok but then E is no longer natural. If you use general continued fractions, for E, Pi and sqrt(2) , well that's better BUT there is gamma which has no representation and we go back to square one. There is no universal answer to this question so far. So far, the real numbers if represented in base 2 for example, is just a huge grey wall with no patterns whatsoever. There is a simple way to get the factorial coeffs. from a number using a greedy algorithm. here is a maple version of this : r2fact:=proc(s) local max, liste, prod, S, T, k; S := evalf(frac(abs(s))); max := 10^(Digits - 2); prod := 1; k := 1; liste := [trunc(s)]; while prod <= max do T := trunc(k!*S); S := S - T/k!; liste := [op(liste), T]; k := k + 1; prod := prod*k end do; RETURN(liste) end proc when used on Pi : 3, 0, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9, 13, 18, 21, 14, 34, 27, 11, 27, 33, ... can you see any pattern in this ?? not me. not gfun either. this is the puzzle number 7514, or if you want : A007514. Best regards, Simon Plouffe
A point plot: http://chesswanks.com/num/a007514.png On Feb 19, 2014, at 9:54 PM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
when used on Pi : 3, 0, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9, 13, 18, 21, 14, 34, 27, 11, 27, 33, ...
can you see any pattern in this ??
#37 and #41 are "wrong". Rich
On Thu, Feb 20, 2014 at 1:30 PM, <rcs@xmission.com> wrote:
#37 and #41 are "wrong".
Here's a direct link to the list of equations he's talking about: http://www.frontiersin.org/Journal/DownloadFile.ashx?sup=1&articleId=74738&F... 37 has prod_{k=0}^oo (1+x^k) = sum_{n=0}^oo p(n) x^n. The LHS should be prod_{k=1}^oo 1/(1-x^k) 41 has n! = n^n e^{-n} sqrt(2 pi n) (1 + o(n)). The RHS should be n^n e^{-n} sqrt(2 pi n) (1 + O(1/n)). -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
An alternative repair for #37 is that the LHS is left alone, with the right-hand side changed from p(n) to q(n), and the description changed to partitions with _unequal_ terms. Quoting Dan Asimov <dasimov@earthlink.net>:
Right, Mike nailed it: 1 + x^k should be 1/(1-x^k). Duh.
On Feb 20, 2014, at 2:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Should #37 have a minus sign, not a plus sign, on the LHS?
(And I had to look up why #41 is wrong. Why is it merely ?wrong? ?) The error term is given as o(n); correct is O(1/n), but o(n) is also true, though not particularly useful. Rich
* rcs@xmission.com <rcs@xmission.com> [Feb 21. 2014 07:59]:
An alternative repair for #37 is that the LHS is left alone,
LHS needs k starting 1 (not 0), no matter what.
with the right-hand side changed from p(n) to q(n), and the description changed to partitions with _unequal_ terms.
Quoting Dan Asimov <dasimov@earthlink.net>:
Right, Mike nailed it: 1 + x^k should be 1/(1-x^k). Duh.
On Feb 20, 2014, at 2:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Should #37 have a minus sign, not a plus sign, on the LHS?
(And I had to look up why #41 is wrong. Why is it merely ?wrong? ?)
The error term is given as o(n); correct is O(1/n), but o(n) is also true, though not particularly useful.
Rich
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Also, #35, #43 (in modern terminology) and #58 are wrong. —Dan
The result is interesting but as George suggests, not surprising. But as a way of rating the equations (Data Sheet 1), it is not a good statistical practice to present the list of equations in the same order to each subject. Also, among the rather small number of subjects (15), some were apparently grad students, who *usually* would not have enough experience with math to appreciate several of the equations. (Present company excluded!) (And one of my favorite equations, the Riemann functional equation for the zeta function — Equation 59 in the list of 60 — is written in perhaps its ugliest possible form, for some inscrutable reason. It’s much nicer when written as xi(s) = xi(1-s), where xi(s) := zeta(s)*gamma(s/2)/pi^(s/2).) Harrumph. —Dan On Feb 19, 2014, at 8:44 AM, George Hart <george@georgehart.com> wrote:
Funsters,
Perhaps not surprising to mathematicians, functional MRI shows that beautiful equations trigger the same physiological response in mathematician's brains as beautiful art or music:
http://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-ac...
George http://georgehart.com/
P.S. My latest Mathematical Impressions video is on the beautiful math/music of change ringing: https://www.simonsfoundation.org/multimedia/mathematical-impressions-change-...
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participants (10)
-
Bill Gosper -
Dan Asimov -
David Makin -
George Hart -
Hans Havermann -
James Buddenhagen -
Joerg Arndt -
Mike Stay -
rcs＠xmission.com -
Simon Plouffe