Minimal surface
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minimal surface
(′min·ə·məl ′sər·fəs)
(mathematics) A surface that has assumed a geometric configuration of least area among those into which it can readily deform.
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Minimal surface
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Minimal surfaces
A branch of mathematics belonging to the calculus of variations, differential geometry, and geometric measure theory. A surface, interface, or membrane is called minimal when it has assumed a geometric configuration of least area among those configurations into which it can readily deform. Soap films spanning wire frames or compound soap bubbles enclosing volumes of trapped air are common examples. See also Differential geometry.
Geometrically, the mean curvature of a surface S at a point is the difference between the maximum upward curvature there and the maximum downward curvature; in particular, a surface of zero mean curvature has such principal curvatures equal and opposite and hence typically appears “saddle-shaped.” It turns out that S is a minimal surface; that is, it cannot be perturbed to less area leaving its boundary fixed, provided the mean curvature is zero at each of its points; such a surface could occur, for example, as a soap film spanning a wire frame. The corresponding minimal surface equation is partial differential equation. If, alternatively, S were part of a soap bubble enclosing trapped air, the mean curvature of S would be proportional to the difference in air pressure between the two sides of S. In the calculus of variations, typically area-minimizing properties of minimal surfaces are emphasized. In differential geometry, minimal surfaces are defined as surfaces of zero mean curvature; surfaces of constant mean curvature are also extensively studied.
The two-dimensional surface is dominant in determining shape whenever the energy of a system is changed significantly by a displacement or a change in area of the surface. Such surfaces include the interfaces between crystals in a typical rock or metal, the film of soapy water between the air cells in a soap froth, the membrane separating the cells in living tissue, and the cracks separating basalt columns. Minimization of surface area plays a role in determining the shape of many living organisms. See also Foam.
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Minimal surface
For minimal surfaces in algebraic geometry see Minimal model (birational geometry).
In mathematics, a minimal surface is a surface with a mean curvature of zero. These include, but are not limited to, surfaces of minimum area subject to various constraints.
Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.
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Examples
Classical examples of minimal surfaces include:
- the plane, which is a trivial case
- catenoids: minimal surfaces made by rotating a catenary once around its symmetry axis
- helicoids: 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
- the Enneper surface
Recent work in minimal surfaces has identified new completely embedded minimal surfaces, that is minimal surfaces which do not intersect. In particular Costa's minimal surface was first described mathematically in 1982 by Celso Costa and later visualized by Jim Hoffman. This was the first such surface to be discovered in over a hundred years. Jim Hoffman, 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.
In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927– ), Robert Longhurst (1949– ), Charles O. Perry (1929–2011), among others.
Definition
Given an embedded surface, or more generally an immersed surface (which may have a fixed boundary, possibly at infinity), one can define its mean curvature, and a minimal surface is one for which the mean curvature vanishes.
The term "minimal surface" is because these surfaces originally arose as surfaces that minimized surface area, subject to some constraint, such as total volume enclosed or a specified boundary, but the term is used more generally.
Minimal surfaces are the critical points for the mean curvature flow: these are both characterized as surfaces with vanishing mean curvature.
The definition of minimal surfaces can be generalized/extended to cover constant mean curvature surfaces: surfaces with a constant mean curvature, which need not equal zero.
A relation to Brownian motion
Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces.[1]
See also
- Bernstein's problem
- Soap bubble
- Plateau's problem
- Stretched grid method
- Curvature
- Weaire–Phelan structure
- Tensile structure
- Enneper–Weierstrass parameterization
- Bilinear interpolation
References
- Robert Osserman (1986). A Survey of Minimal Surfaces. New York: Dover Publications. 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.)
- Hermann Karcher and Konrad Polthier (1995). "Touching Soap Films - An introduction to minimal surfaces". http://page.mi.fu-berlin.de/polthier/booklet/intro.html. Retrieved December 27, 2006. (graphical introduction to minimal surfaces and soap films.)
- Various (2000-). "EG-Models". http://www.eg-models.de/models.html. Retrieved September 28, 2004. (Online journal with several published models of minimal surfaces)
- Stewart Dickson (1996). "Scientific Concretization; Relevance to the Visually Impaired Student". VR in the School, Volume 1, Number 4. http://www.eg-models.de/models.html. Retrieved April 15, 2006. (Describes the discovery of Costa's surface)
- Martin Steffens and Christian Teitzel. "Grape Minimal Surface Library". http://numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/amandus.html. Retrieved October 27, 2008. (An collection of minimal surfaces)
- David Hoffman, Jim Hoffman et al.. "Scientific Graphics Project". http://www.msri.org/about/sgp/jim/geom/minimal/index.html. Retrieved April 24, 2006. (An collection of minimal surfaces with classical and modern examples)
- Jacek Klinowski. "Periodic Minimal Surfaces Gallery". http://www-klinowski.ch.cam.ac.uk/pmsgal1.html. Retrieved February 2, 2009. (An collection of minimal surfaces with classical and modern examples)
- ^ R. Neel 2008 (arXiv, DOI resolver)
External links
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
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