Yesterday, under the subject: Re: [math-fun] analytic function for new Airbnb logo?I blathered ------------ As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg Two more mystery constants associated with this spaceplane: Its volume (4.231765644191597 ± ? (Mma 10.0.2 crashed when I asked for more digits), And its "fattest" (cross-sectional area, vs squarest) q, Out[87]= {q -> 0.43939962899109219485183075448731} which ISC guessed was In[81]:= 1/Log[3]^(7/4)/2^(1/12)/E^(3/5) (nice try). Pix: gosper.org/halphen.pdf The first graph (with strangely varying curvature) is cross sectional area vs q. ---------------------- But just 3.5 years ago (Date 2011-05-10 05:43, Subject Re: [math-fun] Redefine Halphen's constant?) I actually had the bleeping area and volume, and sent ------- 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 ---------------- (ISC: 0 for 2.) And Mma 10 is *still* missing the numerics! In[3]:= D[EllipticTheta[s, t, q], {t, 2}] == -4 q D[EllipticTheta[s, t, q], q] Out[3]= (0,1,0) EllipticThetaPrime [s, t, q] == (0,0,1) -4 q EllipticTheta [s, t, q] In[4] Pi N[% /. t -> -- /. s -> 1 /. q -> 0.69] 4 Out[4]= (0,0,1) 6.91574 == -2.76 EllipticTheta [1., 0.785398, 0.69] Vs my old Macsyma package: (c171) DIFF(THETA[S](T,Q),T,2) = -4*Q*DIFF(THETA[S](T,Q),Q); (d171) thetaderiv (t, 2, q) = - 4 q thetaderiv (t, 0, q, 1) s s (c172) SUBST([T = DFLOAT(%PI/4),S = 1,Q = 0.69],%); (d172) 6.91573338862223d0 = 6.91573338862223d0 Huge thanks to NeilB, Ashok Anumandla, and Babu Lella for (so far, partially) bringing up Macsyma 2.4 under XP under Yosemite! (But I'm still wondering why (d171) no longer auto-simplifies to a tautology.) If it weren't for the missing numerics, Mma would have numerous ways to exactly and manipulably express Halphen's and similar constants: InverseFunction[EllipticTheta^(0,0,1),3,3][1,\[Pi]/4,0] InverseFunction[EllipticTheta^(0,0,1),3,3][2,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][1,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][2,\[Pi]/4,0] Only two of these work: In[311]:= N[%, 22] Out[311]= (0,0,1) {InverseFunction[EllipticTheta , 3, 3][ 1.000000000000000000000, 0.7853981633974483096157, 0] (0,0,1) , InverseFunction[EllipticTheta , 3, 3][ 2.000000000000000000000, 0.7853981633974483096157, 0] , 0.3281065668749782530255, 0.3281065668749782530255} (Notice how it floated the exact subscripts 1 and 2, and now foolishly regards them as approximate.) The mail server's memory is nearly as bad as mine. I recovered the May 2011 text only thanks to a bounce of a copy sent to Steven Finch. More distressing: That mail also contained a bunch of denestings apparently debuted here, but half of them I don't even recognize: Apropos radical denestings, do you think these from math fun startled anybody? 1/5 3/5 3/5 2/5 1/5 2/5 3/5 1/5 1/3 - 2 3 + 2 3 + 3 + 2 (3 - 2 ) = ------------------------------------- 2/3 5 4/5 3/5 2/5 1/5 1/5 2 7 - 7 - 2 7 + 6 7 + 2 sqrt(8 - 7 ) = ----------------------------------- 5 1/5 sqrt(11 76 + 5) = 3/5 4/5 1/5 3/5 4/5 2/5 2/5 1/5 2 19 + 2 19 + 3 2 19 - 2 2 19 - 7 --------------------------------------------------------- 5 6/7 5/7 3/7 1/7 3/7 - 2 + 2 + 2 + 2 2 - 1 sqrt(4 - 1) = --------------------------------- sqrt(7) 5/6 1/6 2/3 1/6 5 2 sqrt(5) sqrt(4 +- sqrt(3) 5 ) = +- --------- -+ ------------ 3 sqrt(2) 3 1/3 5/6 1/6 1/6 5 2 5 2 + ------- +- --------- + ------- sqrt(6) 3 sqrt(3) 1/5 1/5 sqrt(4 - 3 ) = 3/5 4/5 3/5 2/5 2/5 4/5 1/5 1/5 - 2 3 + 3 + 2 2 3 - 2 3 + 2 --------------------------------------------------- 5 Five-termers are quite sparse. I've never been able to find a six-termer. --Bill Wow, who found these? I'd like to meet him! --rwg I just fixed a typo in the Macsyma notebook exported by Mma as http://gosper.org/thetpak.html ,(c575): The last thetaderiv_1 should be thetaderiv_2. This is hard to fix because Mma 10 can no longer read the Mma notebook full of bitmaps pasted in from the Macsyma notebook! (Macsyma never learned how to display formulas in html.) It is a scandal how many thetpak.html examples continue to elude Mma 10. --rwg