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Scherk's Minimal Surfaces

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Scherk's two minimal surfaces were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful images of wood sculptures of Scherk surfaces are illustrated by Séquin.

ScherksMinimalSurface1

Scherk's first surface is doubly periodic and is defined by the implicit equation

e^zcosy=cosx,
(1)

(Osserman 1986, Wells 1991, von Seggern 1993). It has been observed to form in layers of block copolymers (Peterson 1988).

ScherksMinimalSurface2

Scherk's second surface is the surface generated by Enneper-Weierstrass parameterization with

f=4/(1-z^4)
(2)
g=iz.
(3)

It can be written parametrically as

x=2R[ln(1+re^(itheta))-ln(1-re^(itheta))]
(4)
=ln((1+r^2+2rcosphi)/(1+r^2-2rcosphi))
(5)
y=R[4itan^(-1)(re^(itheta))]
(6)
=((1+r^2-2rsinphi)/(1+r^2+2rsinphi))
(7)
z=R{2i(-ln[1-r^2e^(2itheta)]+ln[1+r^2e^(2itheta)])}
(8)
=2tan^(-1)[(2r^2sin(2phi))/(r^4-1)]
(9)

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(4phi))
(10)
F=0
(11)
G=(16r^2(1+r^2)^2)/(1+r^8-2r^4cos(4phi))
(12)

and of the second fundamental form are

e=(8(1+r^4)sin(2phi))/(1+r^8-2r^4cos(4phi))
(13)
f=(8(1-r^4)cos(2phi))/(1+r^8-2r^4cos(4phi))
(14)
g=(8r^2(1+r^4)sin(2phi))/(1+r^8-2r^4cos(4phi)).
(15)

The Gaussian and mean curvatures are

K=-(1+r^8-2r^4cos(4phi))/(4(1+r^2)^4)
(16)
H=0.
(17)

See also

Enneper-Weierstrass Parameterization, 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.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. "Bemerkung über der kleinste Fläche innerhalb gegebener Grenzen." J. reine angew. Math. 13, 185-208, 1834.Séquin, C. H. "Scherk-Collins Sculpture Generator." http://www.cs.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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 223, 1991. Wolfram Research, Inc. "Mathematica Version 2.0 Graphics Gallery." http://library.wolfram.com/infocenter/Demos/4664/.

Cite this as:

Weisstein, Eric W. "Scherk's Minimal Surfaces." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ScherksMinimalSurfaces.html

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