Minimal surface
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Image:Verrill minimal surface.jpg
In mathematics, a minimal surface is a surface with a mean curvature of zero. This includes, but is not limited to, surfaces of minimum area subject to constraints on the location of their boundary.
Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film.
Examples of minimal surfaces include catenoids and helicoids. A minimal surface made by rotating a catenary once around the axis is called a catenoid. A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity is called a helicoid.
Recent work in minimal surfaces has identified new completely embedded minimal surfaces, that is minimal surfaces which do no intersect. In particular Costa's minimal surface was first described mathematically in 1983 by Celsoe Costa and later visualized by Jim Hoffman. This was the first such surface to be discovered in over a hundred years. Jim Hoffmann, David Hoffman and William Meeks III, then extended the definition to produce a family of surfaces with different rotational symmetries.
Minimal surfaces have become an area of intense mathematical and scientific study over the past 15 years, specifically in the areas of molecular engineering and materials science, due to their anticipated nanotechnology applications.
The definition of minimal surfaces can be extended to cover constant mean curvature surfaces.
See also
- soap bubble
- Plateau's problem
- curvature
- Weaire-Phelan structure
- Enneper-Weierstrass parameterization
References
- {{cite book
| author = Robert Osserman | title = A Survey of Minimal Surfaces | year = 1986 | publisher = Dover Publications | location = New York | id = ISBN 0-486-64998-9 }} (Introductory text for surfaces in n-dimensions, including n=3; requires strong calculus abilities but no knowledge of differential geometry.)
- {{cite web
| author = Hermann Karcher and Konrad Polthier | url = http://www.zib.de/polthier/booklet/intro.html | title= Touching Soap Films | year = 2004 | accessdate = September 28 | accessyear = 2004 }} (graphical introduction to minimal surfaces and soap films.)
- {{cite web
| author = Various | url = http://www.eg-models.de/models.html | title= EG-Models | year = 2000- | accessdate = September 28 | accessyear = 2004 }} (Online journal with several published models of minimal surfaces)
- {{cite web
| author = Stewart Dickson | url = http://www.eg-models.de/models.html | title= Scientific Concretization; Relevance to the Visually Impaired Student | year = 1996 | work = VR in the School, Volume 1, Number 4 | accessdate = April 15 | accessyear = 2006 }} (Describes the discovery of Costa's surface)
- {{cite web
| title=Grape Minimal Surface Library | url=http://www.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/amandus.html | author = Martin Steffens and Christian Teitzel | accessdate = April 15 | accessyear = 2006 }} (An collection of minimal surfaces)de:Minimalfläche