A253545 Decimal expansion of r = 0.527697..., a boundary ratio separating catenoid and Goldschmidt solutions in the minimal surface of revolution problem.

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, 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, 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

Offset

0,1

Comments

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

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

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.

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

Links

Table of n, a(n) for n=0..104.

Robert Ferréol's MathCurve, Catenoid

Eric Weisstein's MathWorld, Laplace Limit

Eric Weisstein's MathWorld, Minimal Surface of Revolution

Formula

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

Solution of 2*cosh((x^2+1)/2) = x+1/x. - Robert FERREOL, Feb 07 2019

Example

0.5276973969625715285724233433631805779688537906314195417222751595...

Mathematica

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

Crossrefs

Cf. A033259 (Laplace limit).

Sequence in context: A204899 A354827 A364521 * A195343 A074454 A256110

Adjacent sequences: A253542 A253543 A253544 * A253546 A253547 A253548

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Apr 21 2015

Status

approved