May 2010 - mathfun@mailman.xmission.com

[math-fun] Near closed form for a Somos
by Bill Gosper 27 May '10

27 May '10

Let's call sequences growing like x^n^2 "supergeometric". The Somos recurrence s[n]:=(s[n-1]*s[n-3]+s[n-2]^2)/s[n-4], for s[1],... = 1,1,1,1,..., grows supergeometrically. But if we initialize with 1,1,-1,-2/3,..., we get a supergeometrically *shrinking* sequence = a period 8 sequence times 3^-(n^2/16): s[n] = -2*%i^(n^2+1)*q^(n^2/64)*theta[1](%i*n*log(q)/8+n*%pi/2,q)/ theta[4](0,q)*3^(n^2/16) 2 2 n + 1 n /64 i n log(q) n pi 2 i q theta (---------- + ----, q) 1 8 2 s = - ------------------------------------------ n 2 n /16 theta (0, q) 3 4 = 1,1,-1,-2/3,1/3,1/9,-1/27,0,1/243,-1/729,-1/2187,2/3^9,... where the simplest equation I've found for q is 3^(1/4)*theta[4](0,q) = -theta[2](0,%i*q^(1/4))*%e^(7*i*%pi/8) 7 i pi ------ 1/4 1/4 8 3 theta (0, q) = - theta (0, i q ) e 4 2 Can we do better? Similarly, the period 8 sequence (1,1,-1,0,1,-1,-1,0)* comes from s[n] = theta[1](n*%pi/4,q)/(theta[2](0,q)*%i^(n^2/4-1)) n pi theta (----, q) 1 4 s = -------------------- n 2 n -- - 1 4 theta (0, q) i 2 with q a root of theta[2](0,q) = theta[2](0,%i*q)*sqrt(%i)/sqrt(2) theta (0, i q) sqrt(i) 2 theta (0, q) = ---------------------- 2 sqrt(2) And then there's the even simpler period 5 sequence (1,1,-1,-1,0)* from s[n] = theta[1](2*n*%pi/5,q)/theta[1](%pi/5,q) 2 n pi theta (------, q) 1 5 s = ----------------- n pi theta (--, q) 1 5 with q defined by theta[1](%pi/5,q) = theta[1](2*n*%pi/5,q) pi 2 n pi theta (--, q) = theta (------, q). 1 5 1 5 This theta is presumably algebraic? That would bode ill for closedform q. The only thetas I know how to invert are algebraic times gamma(s). Of course, all of these can be expressed as a pile of trigs But not when we generalize to 1,1,-1,x,... . Can anyone say why this makes polynomials in x? --rwg

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[math-fun] A fan letter to Martin Garner
by bobmayans＠aol.com 26 May '10

26 May '10

I sent this to Martin Gardner, about 25 years ago. -- Bob M. Dear Mr. Gardner, Perhaps you get many letters like this, but I thought I would write anyway, in appreciation of your work in recreational mathematics. I always liked your Mathematical Games column, and as a kid, I would turn to it as soon as my father got Scientific American. In the sixth grade, I made a "hexapawn" playing machine, described in one of these columns, out of empty Benson and Hedges cigarette boxes and colored beads. After getting a degree in mathematics, I started a technical career in computer programming and medical image processing, but missed the fun and brightness of mathematics. I would go to the University of Michigan Library and look over the columns in the bound back issues of Scientific American. It helped to reawaken old interests in the subtlety and beauty of mathematical thought. Later, I got my Ph.D. at Michigan, writing a thesis in set theory, and now I work at Bell Laboratories (alas, not in the mathematics department). I still read your books on mathematical games, every now and then, to remember old pleasures. Sincerely yours, Bob Mayans

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Re: [math-fun] Martin Gardner / G4G9 article at the New Scientist
by Bill Gosper 25 May '10

25 May '10

>In the photo album, image 8 states that "Bill Gosper once held the >record for discovering pi to the greatest number of decimal places." >Wasn't it (greatest number of) terms of its simple continued fraction? Thank you for remembering, Hans! Gawd, I can't live that decimal thing down. The whole point was to try to change the object of the game from decimal to CF. But I needed the decimal too, to compare with Kanada's value. I computed both from an enormous unreduced fraction (binary/binary). Then I computed a reduced fraction from the CF. --rwg

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[math-fun] Quadratic equations --- still getting it wrong!
by Fred lunnon 25 May '10

25 May '10

Several authors have analysed the pitfalls involved in the numerical evaluation of the traditional schoolroom solution of a quadratic equation in one variable x: a x^2 - 2 b x + c = 0 ; x = (b + d)/a , (b - d)/a , where d^2 = b^2 - a c . Very briefly, the relevant considerations are: (1) Rescaling a,b,c, avoiding overflow or underflow in d^2; (2) Double-precision square-root, avoiding cancellation in d^2; (3) Complex roots when d^2 < 0; (4) Evaluating x" = c/(a x') where |x'| > |x"|, avoiding cancellation in x"; (5) Detection of cases a = 0, a = b = 0, a = b = c = 0 (solutions single infinite, double infinite, undefined). The following detailed discussions are freely accessible online: George E. Forsythe "How Do You Solve a Quadratic Equation", Tech. Report CS40, Comp. Sci. Dept., U. of Stanford (1966); Henry G. Baker "Garbage In, Garbage Out" ACM SIGPLAN Notices vol 33, 30--38 (1998). The topic was aired in math-fun a while back ("branchless programming", 22-3/10/2007), but for some reason I didn't at the time get around to contributing my own two-penn'orth. In a geometric context, rather than as a single variable, the solution is often interpreted as a ratio, or homogeneous coordinate (x,y) of some point in projective 1-space. >From this aspect, it's irritating that the traditional formulation ties itself in logical knots over infinite roots (or worse, ignores them altogether); and furthermore it's aesthetically distressing that the symmetry of the homogeneous quadratic form has been lost. But suppose either inhomogeneous solution is written alternatively as x = (b + d)/a = c/(b - d) by symmetry; now adding numerators and denominators leads to the symmetric reformulation: The homogeneous quadratic equation a x^2 - 2 b x y + c y^2 = 0 has solutions x : y = (c + b + d) : (a + b - d) , (c + b - d) : (a + b + d) , where d^2 = b^2 - a c . In this form there is no need to detect and specially treat cases (4) and (5). Where parallel processing is available therefore, it should execute marginally faster than the conventional formula (though of course both versions are dominated by the square root). Fred Lunnon [18/05/10]

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Re: Martin Gardner, October 21, 1914 May 22, 2010
by James Propp 24 May '10

24 May '10

>Sad indeed--just days ago I was proposing writing a paper specifically >because it seemed he'd enjoy it! I myself was in the midst of writing Martin a long-overdue fan letter, explaining the difference he'd made in my life. I finished the letter a week ago, but decided to hold off on sending it until May 28 (so that I could give him the link to a video of a a lecture I'll be giving this Friday). When I made this decision, I even thought to myself "The guy is 95 years old, so there's a chance he could die in the coming week", but I figured it was worth the risk. Now I regret the decision, though it probably wouldn't have made any difference; I don't think he would have been able to read the letter, let alone reply. I suspect now that he was ill for some months -- otherwise I'm sure he would have weighed in with his views on Melanie Bayley's recent claims about there being a secret mathematical agenda in Lewis Carroll's "Alice" books. One way Gardner affected my life is by letting me know that people like Ron Graham, Persi Diaconis, and John Conway existed. They were role models for me before I even met them. And Piet's Hein's grook THE ETERNAL TWINS Taking fun as simply fun and earnestness in earnest shows how thoroughly thou none of the two discernest. has long been a motto of mine. I'd say "So long, Martin, wherever you are!", except that if there's an afterlife with Internet access, I'm sure he's much too busy working out the philosophical impact of life after death to have time for lesser matters. :-) Jim Propp

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[math-fun] iter_exp()
by Steve Witham 24 May '10

24 May '10

I'm (morbidly?) fascinated by iterated exponentials and I like the idea of extrapolating a continuous extension of a function from simple behavior at a limit. The stuff I was doing with sinh and tanh, seemed a bit arbitrary, though. exp() and ln() behave simply near their fixed points. Say this one: ln( z0 ) = z0 = exp( z0 ) ~= 0.318 + 1.337 i Forgive my using the letter d in the following: lim d -> 0 exp( z0 + d ) = z0 + z0 d and so lim d -> 0 exp^n( z0 + d ) = z0 + z0^n d The current place I've wandered to is to define lim n -> oo iter_exp( z ) = exp^n( z0 + z0^(z-n) ) That gives a sort of analogy to e: iter_exp( 0 ) = lim n -> oo exp^n( z0 + z0^-n ) ~= 1.149 + 1.229 i Similar to how exp( 1 ) = lim n -> oo ( 1 + 1/n )^n It looks like a faulty analogy; I haven't come up with a better one so far. The closest thing on Wikipedia is http://en.wikipedia.org/wiki/Tetration#Extension_to_complex_heights which looks very different but I just glanced. Before the picture, other bits: iter_exp( -oo ) = z0 iter_exp( +oo i ) = z0 (Half the plane maps into a disk-like area around the fixed point.) iter_exp( z + 1 ) = exp( iter_exp( z ) ) Which is nice, but there are no z's where iter_exp(z) = 0 or 1 or e or e^e... I may have to learn some complex analysis. Or maybe category theory or LaTeX. Here are pics of a map of the thing before it gets branchy: Two views, each a 75K jpg: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.jpg http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_brain.jpg 395K pdf of the first view: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.pdf Python code: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.py uses this: http://www.mac-guyver.com/switham/2010/03/Superbola/tinyplot.py --Steve

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[math-fun] NYT obit for Martin Gardner
by rcs＠xmission.com 24 May '10

24 May '10

http://www.nytimes.com/2010/05/24/us/24gardner.html :-(

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[math-fun] Pi-calculating engine
by Tom Rokicki 24 May '10

24 May '10

The Pi-calculating Life pattern that the New Scientist mentions was developed by Adam P. Goucher. I'm not positive if this has appeared on math-fun yet, but Andrew J. Wade has built a linear-time replicator in Life: http://pentadecathlon.com/lifeNews/2010/05/oblique_life_spaceship_created.h… -tom -- Check out Golly at http://golly.sf.net/

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[math-fun] Martin Gardner / G4G9 article at the New Scientist
by Thane Plambeck 24 May '10

24 May '10

I count at least three math-fun people (and think I see a fourth) in the photo http://www.newscientist.com/article/dn18950-magic-numbers-a-meeting-of-math… -- Thane Plambeck tplambeck(a)gmail.com http://thaneplambeck.typepad.com/

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[math-fun] gasket addendum
by Bill Gosper 23 May '10

23 May '10

In case you missed it, a fairly slick balanced ternary in Mma: In[1109]:= balter[n_Integer] := Sign[n]*(IntegerDigits[Abs[n] + (3^Ceiling[Log[3, 2*Abs[n]]] - 1)/2, 3] - 1) In[1100]:= balter[322] Out[1100]= {1, 1, 0, 0, -1, 1} Also, from the old mail: [Using fixed-point analysis] I can give exact values of gaskets(t,m) for rational t, and simple formulas when t has a simple description in radix 3. These would make fairly startling identities, especially if we can find a sum or (scalar) product for [A(x)]. pi 2 pi gaskets(- 1, m) = cot(--) - %i, gaskets(- -, m) = cot(--), m 3 m 1 pi %i sqrt(3) pi gaskets(- -, m) = cot(--) - -- + -------, gaskets(0, m) = cot(--) + sqrt(3), 3 m 2 2 m 1 pi %i sqrt(3) 2 pi gaskets(-, m) = cot(--) + -- + -------, gaskets(-, m) = cot(--), 3 m 2 2 3 m pi gaskets(1, m) = cot(--) + %i. m pi For brevity, define S(t) := cot(--) - gaskets(t, m), |t|<=1. Then m S(-t) = conjugate(S(t)) and from the radix 3 expansions of t, 1 3 %i 2 sqrt(3) S(-) = - ---- - ---------, 2 7 7 1 %i 8 sqrt(3) 3 2 %i sqrt(3) S(-) = - -- - ---------, , S(-) = - ---- - -------, 4 5 15 4 5 15 1 %i 4 sqrt(3) 2 3 %i 2 sqrt(3) 3 2 %i sqrt(3) S(-) = - -- - ---------, S(-) = - ---- - ---------, S(-) = - ---- - -------, 5 7 7 5 5 5 5 7 7 4 3 %i S(-) = - ----, 5 5 1 3 %i 9 sqrt(3) 5 9 %i sqrt(3) S(-) = - ---- - ---------, , , , S(-) = - ---- - -------, 6 14 14 6 14 14 1 %i 2 sqrt(3) 2 %i 4 sqrt(3) 3 2 %i sqrt(3) S(-) = - -- - ---------, S(-) = - -- - ---------, S(-) = - ---- - -------, 7 3 3 7 3 7 7 3 3 4 5 %i 4 sqrt(3) 5 %i 6 2 %i sqrt(3) S(-) = - ---- - ---------, S(-) = - --, S(-) = - ---- - -------, 7 21 21 7 3 7 3 7 1 19 %i 176 sqrt(3) 3 38 %i 97 sqrt(3) S(-) = - ----- - -----------, , S(-) = - ----- - ----------, , 8 85 255 8 85 255 5 3 %i 2 sqrt(3) 7 59 %i 44 sqrt(3) S(-) = - ---- - ---------, , S(-) = - ----- - ----------, 8 17 15 8 85 255 1 %i 3 sqrt(3) 2 sqrt(3) 4 3 %i sqrt(3) S(-) = - -- - ---------, S(-) = - -------, , S(-) = - ---- - -------, 9 4 4 9 2 9 4 4 5 %i sqrt(3) 7 %i 8 3 %i sqrt(3) S(-) = - -- - -------, , S(-) = - --, S(-) = - ---- - -------, ... 9 4 4 9 2 9 4 4 --rwg

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