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Scherk's First Surface

ScherksMinimalSurface1

Scherk's first surface is the doubly periodic minimal surface obtained by continuing the graph

z=ln((cosx)/(cosy)),

initially defined for |x|<pi/2 and |y|<pi/2, by half-turn rotations about its straight boundary lines (Scherk 1835, Weber). Its implicit equation is

e^zcosy=cosx.

Its quotient space by the full group of translational periods has genus 0 and four ends (Meeks and Pérez 2012, pp. 25-26).

The surface has been observed in layers of block copolymers (Peterson 1988, Thomas et al. 1988). It belongs to Scherk's minimal surfaces.

See also

Doubly Periodic Minimal Surface, Minimal Surface, Scherk's Minimal Surfaces, Scherk's Second 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 First Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScherksFirstSurface.html

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