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Scherk's Second Surface

ScherksMinimalSurface2

Scherk's second surface is a singly periodic minimal surface generated by Enneper-Weierstrass parameterization with

f=4/(1-z^4)
(1)
g=iz.
(2)

It can be written parametrically as

x=2Re[ln(1+re^(itheta))-ln(1-re^(itheta))]
(3)
=ln((1+r^2+2rcostheta)/(1+r^2-2rcostheta))
(4)
y=Re[4itan^(-1)(re^(itheta))]
(5)
=ln((1+r^2-2rsintheta)/(1+r^2+2rsintheta))
(6)
z=Re{2i(-ln[1-r^2e^(2itheta)]+ln[1+r^2e^(2itheta)])}
(7)
=2tan^(-1)[(2r^2sin(2theta))/(r^4-1)]
(8)

for theta in [0,2pi), and r in (0,1). With this parametrization, the coefficients of the first fundamental form are

E=(16(1+r^2)^2)/(1+r^8-2r^4cos(4theta))
(9)
F=0
(10)
G=(16r^2(1+r^2)^2)/(1+r^8-2r^4cos(4theta)),
(11)

and those of the second fundamental form, writing X(r,theta)=(x,y,z) and using the normal in the direction of X_r×X_theta, are

e=(8(1+r^4)sin(2theta))/(1+r^8-2r^4cos(4theta))
(12)
f=(8r(1-r^4)cos(2theta))/(1+r^8-2r^4cos(4theta))
(13)
g=-(8r^2(1+r^4)sin(2theta))/(1+r^8-2r^4cos(4theta)).
(14)

The Gaussian curvature and mean curvature are

K=-(1+r^8-2r^4cos(4theta))/(4(1+r^2)^4)
(15)
H=0.
(16)

The corresponding implicit equation is

sin(z/2)=sinh(x/2)sinh(y/2).
(17)

The given parameterization describes a fundamental patch. Continuation gives the complete singly periodic minimal surface, with translational period 4pi in the z-direction in this normalization. It belongs to Scherk's minimal surfaces.

See also

Enneper-Weierstrass Parameterization, Minimal Surface, Scherk's First Surface, Scherk's Minimal Surfaces, Singly Periodic Minimal Surface

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References

Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40, 1990.do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 41, 1986.Meeks, W. H., III and Pérez, J. A Survey on Classical Minimal Surface Theory. Providence, RI: Amer. Math. Soc., 2012. https://doi.org/10.1090/ulect/060.Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 18 and 101, 1986.Peterson, I. "Geometry for Segregating Polymers." Sci. News 134, 151, Sep. 3, 1988.Scherk, H. F. "Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen." J. reine angew. Math. 13, 185-208, 1835. https://doi.org/10.1515/crll.1835.13.185.Séquin, C. H. "Scherk-Collins Sculpture Generator." https://people.eecs.berkeley.edu/~sequin/SCULPTS/scherk.html.Thomas, E. L.; Anderson, D. M.; Henkee, C. S.; and Hoffman, D. "Periodic Area-Minimizing Surfaces in Block Copolymers." Nature 334, 598-601, 1988.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993.Weber, M. "Scherk's Doubly Periodic Surface." https://minimalsurfaces.blog/home/repository/doubly-periodic/scherks-doubly-periodic-surface/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 223, 1991. Wolfram Research, Inc. "Mathematica Version 2.0 Graphics Gallery." https://library.wolfram.com/infocenter/Demos/4664/.

Cite this as:

Weisstein, Eric W. "Scherk's Second Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScherksSecondSurface.html

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