For some reason, the original message
From rwg@sdf.lonestar.org Mon Aug 18 05:49:22 2008 is missing from my inbox, but the followup is there, and both are in the archive, so hopefully the rest of math-fun got both.
These were about the family of closed curves described parametrically by (theta_2,theta_1}, which if 3D plotted for 0<q<.9, draws a "space plane" whose fuselage cross sections range from a tiny circle through a near square to nearly a plus-sign. ParametricPlot3D[{EllipticTheta[1, t, q], EllipticTheta[2, t, q], q}, {t, 0, 2*\[Pi]}, {q, 0, .9}, BoxRatios -> {1, 1, 1}] The q values maximizing area and "squareness" are respectively ParametricPlot[{{EllipticTheta[1,t, .4393996289909801220193544165615151], EllipticTheta[2, t, .4393996289909801220193544165615151]}, {EllipticTheta[1, t, .32810656687497825302548], EllipticTheta[2, t, .32810656687497825302548]}}, {t, 0, 2*Pi}] What's new tonight is an exact formula for the cross-sectional area 2 Pi EllipticThetaPrime[1, 0, q^2] (which can be written in Macsyma as 4 * pi * eta^3(q^4) ) and volume 2 pi^2 * sech(pi/sqrt(2)) . Before I found the area formula, Julian labored mightily to get 30 digits of the q maximizing area, difficult because Mma 8.0 seems to be missing the numerics: In[1854]:= Derivative[0, 1, 1][EllipticTheta][1, 69, .105] Out[1854]= (0,1,1) EllipticTheta [1, 69, 0.105] I'm puzzled by this apparently nondescript q and the corresponding maximal shape. I found no relations other than the Halphen's result . --rwg Cc: rwg@sdf.lonestar.org <I should have gotten two!> Subject: [math-fun] Redefine Halphen's constant? X-List-Received-Date: Mon, 18 Aug 2008 11:49:22 -0000 If you try to approximate e^-x on [0,oo) with a ratio of polynomials of degree n, then instead of vanishing, the error for large x will approach the ratio of the leading coefficients, which will also be the ripple height in the best (minimax) approximation. For successively larger n, this ripple height goes down by a factor which was erroneously conjectured to be 1/9, but is more like .107653919, and is somewhat wryly called the One Ninth constant. But it is sometimes also called Halphen's constant, because its square root coincidentally solves the equation 0=Theta_2''(0,i q), posed by Halphen. This square root (.32810656687497825302548) also has nice geometric properties w.r.t. the family of curves (or surface) whose parametric equation is (x,y) = (Theta_2(t), Theta_1(t)), 0<q<1. For small q this is a tiny "circle" which enlarges and grows "squarer" as q -> .328, in the manner of |x|^n + |y|^n = r^n, but rotated by pi/4, so that the incircle is tangent at +-x = +-y (t= (2k+1) pi/4). At q=.328, the incircle is maximal (r = 1.34883916748787) and the curvature vanishes at the four tangencies. The "CRT screen" exponent is n=5.419, a very close fit, but x^n+y^n has nonvanishing curvature at the axes. For q>.328, the curve develops concavities, eventually approaching an infinite "plus" sign. The equation stating the vanishing curvature at t=pi/4 is Theta_2'(pi/4) Theta_1''(pi/4) = Theta_1'(pi/4) Theta_2''(pi/4). But Theta_2'(pi/4) = Theta_1'(pi/4), and obviously doesn't vanish, and since Theta_2''(pi/4) = Theta_1''(pi/4) for all q, the Theta'' must vanish on both sides of the eqn. (I.e., the parametric curves must have their inflections at pi/4.) Maximality of the inradius is d Theta_2(pi/4)/dq = 0, but d Theta_s(z)/dq = -Theta_s''(z)/4/q, which for z=pi/4 is equivalent to the vanishing curvature. Alternative expressions for d Theta_2(pi/4)/dq are d Theta_2(0,i q)/dq /sqrt(2)/i^(1/4) = - Theta_2''(0,i q)/4/sqrt(2)/i^(1/4)/q = - Theta_1''(pi/4,q)/4/q . The area is maximal (7.40051953747301) at q = 0.43939962899098. Adding a z coordinate of q produces a surface resembling a squat, blunt-nosed bomb with four infinitely spreading tailfins, volume = 4.231765651557. Anyway, howsabout we honor Halphen with .328 and relieve "One ninth" of its extra, though more decorous synonym? Plouffe's Inverter might be the most influential current reference using "Halphen's" to mean "One ninth", which is (fortunately) discouraged in the current edition of Finch's (more influential) Constants book. The Weissopedia sez "See One Ninth constant", but doesn't come right out and say they're the same. --rwg (This exercise exposed many deficiencies in my Macsyma Theta function package (http://www.tweedledum.com/rwg/thet.gif, not with I.E.), but surprisingly more in Mma 6.0.) Great, politically incorrect Mythbusters question: Is it actually possible to break a (healthy) camel's back with straw?
From rwg@sdf.lonestar.org Tue Aug 19 06:22:27 2008 From: rwg@sdf.lonestar.org To: rwg@sdf.lonestar.org User-Agent: SquirrelMail/1.4.9a Cc: math-fun@mailman.xmission.com, rwg@sdf.lonestar.org Subject: Re: [math-fun] Redefine Halphen's constant? X-List-Received-Date: Tue, 19 Aug 2008 12:22:27 -0000
Adding a z coordinate of q produces a surface resembling a squat, blunt-nosed bomb with four infinitely spreading tailfins, volume = 4.231765651557.
Actually, it looks rather hypersonic if you crop the infinite tail and stretch it a bunch: http://gosper.org/sst.png .
volume = 4.231765651557. Per Macsyma. Mma is still working on the twelve digits I asked for yesterday.
This exercise exposed many deficiencies in my Macsyma Theta function package
Mostly in the area of simplifying special values and derivatives. Analogous to trig (under control of the same (%piargs) switch), it now translates out quarterperiods and quarterquasiperiods (which can get messy): [...] <Let me know if you want them re-sent.> [...] %pi 3 8 thetaderiv (---, 1, q) = 4 eta (q ) 4 4 The 9th derivative wrt q at q=0 of the nth derivative wrt z of theta[3](z,q): (c436) thetaderiv[3](z,n,0,9) n %pi n (d436) 725760 6 cos(6 z + -----) 2
I wonder if WRI would like one of these.-) --rwg