October 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Another approximation of pi
by David Wilson 14 Oct '12

14 Oct '12

On 10/13/2012 3:19 AM, Peter Luschny wrote: > Alonso del Arte: >> As far as I know, our own Daniel Forgues is the first to notice that >> sqrt(9.87654321) = 3.1426968... which is about as good an approximation of >> pi as 22/7. For some Sequences of the Day in September, to suggest that a >> keyword:cons sequence ought to be chosen, I put in the number 9.87654321 >> purely as a placeholder. To my pleasant surprise, Dan added his observation >> to the September 30 entry. > PL> Well, the problem here is that you did not stop your placeholder at 9.87. > PL> This would have given Daniel the chance to notice a much better > PL> approximation of Pi than 22/7 > > DW> Hence pi ~= sqrt(9.87654321) ~= sqrt(200/81) = 20 sqrt(2) / 9. sqrt(200/81) should have been sqrt(800/81), sorry. > > Now what's the point of this? > > 20*sqrt(2)/9 - Pi = .0011041516837513144 > sqrt(9.87) - Pi = .0000629608912084013 > > Peter > What's the point of any of it? Clearly sqrt(9.87) beats sqrt(9.87654321) in accuracy, but there's nothing particularly exciting about sqrt(9.87), If I was concerned only with accuracy, I would proffer the yet more accurate and uninteresting sqrt(9.8696). If any of these expressions had a practical use, it would be in reducing manual computations, but here sqrt(9.87) is inferior to 355/133 or even 3.1416 in both accuracy and computation facility. Now that computers have largely obsoleted manual computations, the only value of this sort of approximation is curiosity, and I found it curious that 9.87654321 ~= 800/81 led to an expression in which so many squares dropped out of the radical.

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Re: [math-fun] Another approximation of pi
by Adam P. Goucher 14 Oct '12

14 Oct '12

And all of them pale in comparison with Ramanujan's ln(640320^3+744)/sqrt(163). (Actually, there's some sum which approximates pi/8 to over 40 digits of accuracy. But that's cheating slightly, using an infinite series instead of a closed form.) Sincerely, Adam P. Goucher http://cp4space.wordpress.com > ----- Original Message ----- > From: Eugene Salamin > Sent: 10/13/12 09:32 PM > To: math-fun > Subject: Re: [math-fun] Another approximation of pi > > A much more interesting approximation is Ramanujan's π = (2143/22)^(1/4), good to 9 places. > > > -- Gene > > > > > >________________________________ > > From: David Wilson <davidwwilson(a)comcast.net> > >To: Math Fun <math-fun(a)mailman.xmission.com> > >Sent: Saturday, October 13, 2012 11:43 AM > >Subject: Re: [math-fun] Another approximation of pi > > > >On 10/13/2012 3:19 AM, Peter Luschny wrote: > >> Alonso del Arte: > >>> As far as I know, our own Daniel Forgues is the first to notice that > >>> sqrt(9.87654321) = 3.1426968... which is about as good an approximation of > >>> pi as 22/7. For some Sequences of the Day in September, to suggest that a > >>> keyword:cons sequence ought to be chosen, I put in the number 9.87654321 > >>> purely as a placeholder. To my pleasant surprise, Dan added his observation > >>> to the September 30 entry. > >> PL> Well, the problem here is that you did not stop your placeholder at 9.87. > >> PL> This would have given Daniel the chance to notice a much better > >> PL> approximation of Pi than 22/7 > >> > >> DW> Hence pi ~= sqrt(9.87654321) ~= sqrt(200/81) = 20 sqrt(2) / 9. > >sqrt(200/81) should have been sqrt(800/81), sorry. > >> > >> Now what's the point of this? > >> > >> 20*sqrt(2)/9 - Pi = .0011041516837513144 > >> sqrt(9.87) - Pi = .0000629608912084013 > >> > >> Peter > >> > >What's the point of any of it? Clearly sqrt(9.87) beats sqrt(9.87654321) in accuracy, but there's nothing particularly exciting about sqrt(9.87), If I was concerned only with accuracy, I would proffer the yet more accurate and uninteresting sqrt(9.8696). If any of these expressions had a practical use, it would be in reducing manual computations, but here sqrt(9.87) is inferior to 355/133 or even 3.1416 in both accuracy and computation facility. Now that computers have largely obsoleted manual computations, the only value of this sort of approximation is curiosity, and I found it curious that 9.87654321 ~= 800/81 led to an expression in which so many squares dropped out of the radical. > > > > > > > > > >_______________________________________________ > >math-fun mailing list > >math-fun(a)mailman.xmission.com > >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Faster than light?
by Warren Smith 13 Oct '12

13 Oct '12

(Cloos is just wrong: massless particles with energy are well known, photons are examples. Also, some previous poster was just wrong in asserting that while in QED electrons could be interpreted as positrons moving backwards in time, and Feynman diagrams included "backwards causal" components.... this was no longer true in electroweak theory. Nonsense, it is true in every such theory. However, there are ways of interpreting both QED and those other theories in ways which "save causality." To explain that in an oversimplified way, you basically, whenever a particle moves backward in time, or moves faster than light -- you could instead regard it as its antiparticle moving forward in time. Thus causality and the existence of antimatter are inextricably interlinked. Furthermore there are ways to argue that all the remaining acausal effects, arising from faster than light motion, magically cancel whenever one computes anything that actually could be measured, because then both these plain-particle and anti-particle interpretations happen and yield canceling effects. It's quite magical and is discussed in Peskin & Schroeder near the start of the book.) ===== Perhaps more interesting that my original objection about information transmission using these tachyons, is the fact that presumably we could apparently-violate conservation of energy & momentum too. That is: 1. You emit photon. 2. photon interacts with tachyon. 3. You receive photon back again. During step 2, photon mysteriously (if you did not know about tachyons) loses or gains energy and/or momentum. The thing about tachyons is they need to be invisible and incapable of interacting with normal particles, or we get paradoxes. But if they can interact with photons we can effectively "see" them... ditto gravitons... and so I don't see how they avoid paradoxes both informational and in terms of apparent conservation violations. Somebody should actually read their paper... Proc. Royal Soc. A 2012...

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[math-fun] faster than light?
by Warren Smith 13 Oct '12

13 Oct '12

Here is a press report http://phys.org/news/2012-10-physicists-special-relativity.html#jCp on paper discussing new way to treat special relativity permitting faster than light particles (tachyons). The momentum of an infinite-velocity particle is in this picture a finite and nonzero consnt depending only on the particle-type. Apparently then you can get conserved real momentum and energy and these kinds of particle could only travel faster than light, never below. I have not yet read the paper itself (behind paywall). It seems to me, though, that such tachyons could interact with exactly-lightspeed=massless particles like photons & gravitons. (Also, e.g, W-bosons have zero mass until they gain effective mass from the Higgs mechanism.) Those in turn could interact with normal particles. Thus effectively tachyons & normals could interact. That would permit transmission of information from normal matter into the past of normal matter, creating paradoxes. If such paradoxes are considered forbidden, we'd presumably still need to conclude that tachyons cannot exist.

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[math-fun] Helical gliders and continuous cellular automata
by Adam P. Goucher 12 Oct '12

12 Oct '12

Tim Hutton, Robert Munafo and some other people have been experimenting with continuous cellular automata and partial differential equations in two and three dimensions. http://cp4space.wordpress.com/2012/10/12/spinning-around/ Sincerely, Adam P. Goucher

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[math-fun] ASCII Typesetting complaint
by Warren Smith 11 Oct '12

11 Oct '12

>cwg: Warren, you have GMail. Neil just taught me to click on?, then "Message text garbled?" --actually there is no "?" button for me to click. Nor is there a "Message text garbled?" button. >lunnon: Simply select View Source or Show Original options on the message window, and the uncorrupted message should be displayed as a textfile, with correct spacing and newlines. --actually there is no "Show Original". There is "View Source"... HOORAY, THAT ACTUALLY WORKED! course you have to then see a heck of a lot of other garbage, but ok. Thank you.

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[math-fun] Roger-Ramanujan as matrix product for contfrac
by Joerg Arndt 10 Oct '12

10 Oct '12

As I suspected the matrix product for contfracs is helpful: Write convergents P_k, Q_k of a0 + b1/(a1 + b2/(a2 + b3/(a3 + ... ))) as matrix product: > [ P P ] [ P P ] [ a 1 ] ___k [ a 1 ] > [ k k-1 ] [ k-1 k-2 ] [ k ] | | [ j ] > [ ] = [ ] [ ] = | | [ ] > [ Q Q ] [ Q Q ] [ b 0 ] | |j=0 [ b 0 ] > [ k k-1 ] [ k-1 k-2 ] [ k ] [ j ] > (fxtbook p.720 gives transposed version). Now we have (set b_k = a*q^k and a_k = 1) > inf > /===\ > | | [ 1 1 ] [ V V ] (for a==1) [ G G ] > | | [ ] = [ ] = [ ] > k = 0 [ a*q^k 0 ] [ U U ] [ H G ] where G(q)=sum(n=0,N,q^(n^2)/prod(k=1,n,1-q^k)); /* g.f. of A003114 */ H(q)=sum(n=0,N,q^(n^2+n)/prod(k=1,n,1-q^k)); /* g.f. of A003106 */ U(a,q) = a * sum(n=0,N, a^n*q^(n^2+n) / prod(k=1,n, 1-q^k) ); \\ = H(q) for a==1 \\ Note I kept factor a for consistency with prior postings V(a,q) = sum(n=0,N, a^n*q^(n^2) / prod(k=1,n, 1-q^k) ); \\ = G(q) for a==1 V(a,q) / U(a,q) = 1/a * ( 1+a*q/(1+a*q^2/(1+a*q^3/(1+a*q^4/(1+a*q^5/(1+...))))) ) (We certainly can get rid of the ugly factor 1/a, but my brain is not working anymore today.) And for a==1 this is the Roger-Ramanujan contfrac G(q) / H(q) = 1+q/(1+q^2/(1+q^3/(1+q^4/(1+...)))) see https://oeis.org/A003823 Now it's officially beer o'clock.

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[math-fun] tiny 2x2 simultaneously computes both Rogers-Ramanujan sums
by Bill Gosper 10 Oct '12

10 Oct '12

(For R-R, special-case a:=q) (c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]); [ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] (c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6); [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] (d45)/T/ [ ] = [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] [ ] [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] This came from the eensy path-invariant 3x3s {{k, {{0, q^n, 0}, {q^(k + n), q^n, 1 - q^n}, {0, 0, 1}}}, {n, {{q^(k + 2 n)/(1 - q^(1 + n)), 0, q^n}, {0, q^(k + 2 n)/(1 - q^(1 + n)), 1}, {0, 0, 1}}}}. --rwg with the usual big help from Corey&Julian

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[math-fun] ASCII Typesetting complaint
by Warren Smith 10 Oct '12

10 Oct '12

Cannot you, if you insist on ASCII-art typesetting, tell your typesetting program the screenwidth is some small-enough number of characters, so that the mailer or whatever will not wrap? Basically, everything Gosper says is just garbage characters when I try to read it. Sure, if I care enough, I can probably reconstruct, but really this is asking a lot of the poor reader that not only is he expected to read Gosper's stuff, which is hard enough even if it really were beautifully typeset, but he also is expected to decrypt it.

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Re: [math-fun] Two twisted cubes
by Adam P. Goucher 09 Oct '12

09 Oct '12

We can generalise this to other regular polytopes. Regular simplexes are pretty triv: f(2) = arccos(1/2) = pi/3 f(3) = arccos(1/3) f(4) = arccos(1/4) ... Sincerely, Adam P. Goucher http://cp4space.wordpress.com ----- Original Message ----- From: Warren Smith Sent: 10/09/12 11:02 PM To: math-fun Subject: [math-fun] Two twisted cubes Consider two 0-centered N-dimensional hypercubes, rotated with respect to each other. Let f(N) = max(over all rotations) min(over all red-blue vertex pairs AB) angle(A0B). For example, f(1)=0, f(2)=45 degrees. This function f is an interesting geometrical quantity. How does this sequence continue? What are its asymptotics? Does f(N) --> 0 when N-->infinity? Or to some positive constant (what)? I know f(4)>=45 degrees. I can prove f(N) > C for some positive constant C (which I could compute) for all large-enough N. I have no upper bound of the form f(N) < C < 90 degrees, but surely one must be true. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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