Gabe: I doubt you've played around with EllipticE, but do you have a prediction on whether there is a smallest positive eccentricity with perimeter expressible with Gammas (Factorials)? Or is there an endless progression of rounder and rounder ellipses with such perimeters?

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I hope your mathematical intuition favored the latter! How weird would

it be for some nondescript algebraic number to be *the* smallest positive eccentricity with a perimeter expressible in Gammas? Check out this one! The perimeter formula is 4 EllipticE[eccentricity^2=1-semiminor axis^2] (Where major axis is scaled to 2): Out[519]= 4 EllipticE[1/(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4] == perim[Sqrt[1 - 1/(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4]] = perim[minor axis/major axis] = perim[√(1 - eccentricity^2)], where the eccentricity = In[520]:= e == √%[[1, 2, 1]] Out[520]= e == 1/(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^2 In[522]:= N[{%520, %519}] Out[522]= {e == 0.0173323801209993, 6.28271339547164 == perim[0.9998497830171]} where a perfect circle has eccentricity 0 and perim[1] = 2 π = 6.2831... . Out[527]= perim[Sqrt[1 - 1/(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4]] == 4 EllipticE[1/(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4] == ( 4 √π (((1 + Sqrt[3/2] + 1/Sqrt[2] + 1/Sqrt[3]) Gamma[1/3])/Gamma[5/6] + (2 Gamma[5/6])/Gamma[1/3]))/( 3^(1/4) (4 + Sqrt[2] + Sqrt[6])) In[529]:= N[List @@ %527] Out[529]= {perim[0.9998497830171], 6.28271339547164, 6.28271339547164} (I.e., the EllipticE[...^-4] = a big pile of Gammas very nearly π/2 = EllipticE@0) The endless crawl toward circularity follows from recurrent identities ("downward transformations") relating Elliptick and EllipticE of successively smaller arguments: EllipticK[k] == (2 EllipticK[(1 - Sqrt[1 - k])^2/(1 + Sqrt[1 - k])^2])/(1 + Sqrt[1 - k]) EllipticE[k] == (1 + Sqrt[1 - k]) EllipticE[(1 - Sqrt[1 - k])^2/(1 + Sqrt[1 - k])^2] - Sqrt[1 - k] EllipticK[k] (Borwein & Borwein, p 12.) WARNING: This is Mathematica's modern notation, where k is the "parameter", not the Borwein's old "modulus" notation! parameter = modulus^2. parameter' := 1 - parameter, modulus' := √(1-modulus²). Given the headaches this cost, I'm surprised it ever got fixed. A similar upheaval "fixed" the Bernoulli numbers. Sadly, they're still messed up: BernoulliB[n] should be BernoulliB[n,1], not BernoulliB[n,0]. —Bill Ask me for the whole Borwein & Borwein pdf, "Pi and the AGM", 414pp.