

A253545


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


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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
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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 socalled Goldschmidt discontinuous solution consisting of the 2 disks.
If d/D > L, there remains only the Goldschmidt solution.


LINKS



FORMULA

arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^21) + arccosh(u)  u^2 = 0.


EXAMPLE

0.5276973969625715285724233433631805779688537906314195417222751595...


MATHEMATICA

digits = 105; u0 = u /. FindRoot[u*Sqrt[u^21] + ArcCosh[u]  u^2 == 0, {u, 6/5}, WorkingPrecision > digits+5]; r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First


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KEYWORD



AUTHOR



STATUS

approved



