Scherk's Minimal Surfaces

A class of Minimal Surfaces discovered by Scherk (1834) which were the first new surfaces discovered since Meusnier in 1776. Scherk's first surface is doubly periodic. Scherk's second surface, illustrated above, can be written parametrically as

$\displaystyle x$	$\textstyle =$	$\displaystyle 2\Re[\ln(1+re^{i\theta})-\ln(1 - re^{i\theta})]$
$\displaystyle y$	$\textstyle =$	$\displaystyle \Re[4i\tan^{-1}(re^{i\theta})]$
$\displaystyle z$	$\textstyle =$	$\displaystyle \Re\left\{{2i(-\ln[1 - r^2 e^{2i\theta}] + \ln[1 + r^2e^{2i\theta}])}\right\}$

for $\theta\in [0,2\pi)$ , and $r\in(0,1)$ . Scherk's first surface has been observed to form in layers of block copolymers (Peterson 1988).

von Seggern (1993) calls

$\begin{displaymath} z=c\ln\left[{\cos(2\pi y)\over\cos(2\pi x)}\right] \end{displaymath}$

``Scherk's surface.'' Beautiful images of wood sculptures of Scherk surfaces are illustrated by Séquin.

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.

Peterson, I. ``Geometry for Segregating Polymers.'' Sci. News, 151, Sep. 3, 1988.

Scherk, H. F. ``Bemerkung über der kleinste Fläche innerhalb gegebener Grenzen.'' J. Reine. angew. Math. 13, 185-208, 1834.

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.

Wolfram Research. ``Mathematica Version 2.0 Graphics Gallery.'' http://www.mathsource.com/cgi-bin/MathSource/Applications/Graphics/3D/0207-155.