Touching Soap Films

An Introduction to Minimal Surfaces
By Hermann Karcher and Konrad Polthier

Introduction
Plateau
History
Visualization
Architecture
Crystallography
Weierstraß
Properties 1
Properties 2
Properties 3
Symmetry
Alteration
Periodic
Handles
Production
Scenes 1
Scenes 2
Scenes 3
Results
Exhibition
Numerics
References
Web Links

References and Further Reading

All images in this booklet are copyright protected. The images are taken from the video except figure 4 which is used with kind permission of the Institut für Leichte Flächentragwerke, Universität Stuttgart, and figure 6 taken from H.A. Schwarz Gesammelte Mathematische Abhandlungen, Springer Verlag, Berlin 1890.

Hermann Karcher and Konrad Polthier Construction of Triply Periodic Minimal Surfaces in: Phil. Trans. R. Soc. Lond. A, vol. 354, pp. 2077-2104 (1996).

Hermann Karcher and Konrad Polthier Die Geometrie von Minimalflächen. Spektrum der Wissenschaft, vol. 10, pp. 96-107 (1990) - Reprinted in: G. Faltings (Ed.) Moderne Mathematik Spektrum Verlag 1996.

Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, Ortwin Wohlrab Minimal Surfaces (2 volumes) Springer Verlag 1992.

Johannes C. C. Nitsche Lectures on Minimal Surfaces vol I and II, Cambridge University Press 1989.

Gerd Fischer Mathematische Modelle (2 volumes), Akademie-Verlag, Berlin 1986.

Stefan Hildebrandt and Anthony Tromba Parsimoneous Universe Springer Verlag Heidelberg.

David Hoffman The Computer-Aided Discovery of New Embedded Minimal Surfaces. The Mathematical Intelligencer, vol. 9, no. 3, pp. 8-21 (1987).

Sten Andersson, S. T. Hyde, K. Larsson, and S. Lidin Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers. Chemical Reviews, vol. 88, pp. 221-242 (1988).

Hermann Karcher The Triply Periodic Minimal Surfaces of Alan Schoen and Their Constant Mean Curvature Companions. Manuscripta Mathematica, vol. 64, pp. 191-357 (1989).

Hermann Karcher Embedded Minimal Surfaces Derived from Scherk's Examples. Manuscripta Mathematica, vol. 62, pp. 83-114 (1988).

Robert Osserman A Survey of Minimal Surfaces Van Nostrand New York, 1969.

Konrad Polthier New Minimal Surfaces in H³ in: Theoretical and Numerical Aspects of Geometric Variational Problems, G. Dziuk, G. Huisken, J. Hutchinson (Eds.), Proc. of the CMA at ANU Canberra vol 26, 1991.

Konrad Polthier Geometric A Priori Estimates For Hyperbolic Minimal Surfaces. Bonner Mathematische Schriften, No. 263, 1994.

Ulrich Pinkall and Konrad Polthier Computing Discrete Minimal Surfaces and Their Conjugates. Experimental Mathematics, Vol 2 (1), 1993, pp. 15-36.

Harold Rosenberg Some Recent Developments in the Theory of Properly Embedded Minimal Surfaces in R³. Séminaire Bourbaki 44ème année, 1991-92, no 759.

Konrad Polthier and Martin Rumpf A Concept For Time-Dependent Processes in: Visualization in Scientific Computing. Eds: M. Göbel, H. Müller, B. Urban pp. 137-153 Springer Verlag 1995.