A253545 - OEIS

%I #27 Feb 07 2019 15:42:08

%S 5,2,7,6,9,7,3,9,6,9,6,2,5,7,1,5,2,8,5,7,2,4,2,3,3,4,3,3,6,3,1,8,0,5,

%T 7,7,9,6,8,8,5,3,7,9,0,6,3,1,4,1,9,5,4,1,7,2,2,2,7,5,1,5,9,5,0,1,6,2,

%U 0,7,6,8,3,2,4,5,1,9,8,8,4,4,6,6,8,4,5,2,9,3,6,0,0,5,4,7,5,3,0,3,5,1,4,1,5

%N Decimal expansion of r = 0.527697..., a boundary ratio separating catenoid and Goldschmidt solutions in the minimal surface of revolution problem.

%C Consider two circular frames each of diameter D and with a separation of d.

%C If d/D < r = 0.527697..., then a catenoid gives the absolute minimum area.

%C If r < d/D < L = 0.66274... (Laplace limit), there are 3 minimal surfaces of revolution passing through the frames: 2 catenoids and the so-called Goldschmidt discontinuous solution consisting of the 2 disks.

%C If d/D > L, there remains only the Goldschmidt solution.

%H Robert Ferréol's MathCurve, <a href="http://www.mathcurve.com/surfaces.gb/catenoid/catenoid.shtml">Catenoid</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/MinimalSurfaceofRevolution.html">Minimal Surface of Revolution</a>

%F arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^2-1) + arccosh(u) - u^2 = 0.

%F Solution of 2*cosh((x^2+1)/2) = x+1/x. - _Robert FERREOL_, Feb 07 2019

%e 0.5276973969625715285724233433631805779688537906314195417222751595...

%t digits = 105; u0 = u /. FindRoot[u*Sqrt[u^2-1] + ArcCosh[u] - u^2 == 0, {u, 6/5}, WorkingPrecision -> digits+5];  r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First

%Y Cf. A033259 (Laplace limit).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 21 2015