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

Author

Jean-François Alcover, Apr 21 2015

Keywords

• nonn
• cons
• easy

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.

Examples

			0.5276973969625715285724233433631805779688537906314195417222751595...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Robert Ferréol's MathCurve, Catenoid
• Eric Weisstein's MathWorld, Laplace Limit
• Eric Weisstein's MathWorld, Minimal Surface of Revolution

Crossrefs

Cf. A033259 (Laplace limit), A202357.

Programs

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

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

Equals sqrt(A202357). - Hugo Pfoertner, Dec 21 2024