Verrill Minimal Surface

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. Verrill Minimal Surface, generated by Eramesan using MATLAB File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Verrill Minimal Surface, generated by Eramesan using MATLAB File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, mean curvature of a surface is a notion from differential geometry. ...

Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film. A soap film is a physical realization of a minimal surface. ...

Examples of minimal surfaces include catenoids, helicoids and the Enneper surface. 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. A catenoid A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. ... The helicoid is one of the first minimal surfaces discovered. ... In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is described parametrically by the following set of equations: It was introduced by Alfred Enneper in connection with minimal surface theory. ...

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. A rendering of Costas minimal surface. ... Jim Hoffman is a software engineer based in Alameda, California, who has worked in mathematical visualization and produced the first visualization of Costas minimal surface. ...

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. This article needs to be cleaned up to conform to a higher standard of quality. ... The Materials Science Tetrahedron, which often also includes Characterization at the center Materials science is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. ... Molecular gears from a NASA computer simulation. ...

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
• Tensile architecture
• Enneper-Weierstrass parameterization

A soap bubble A soap bubble is a very thin film of soap water that forms a hollow sphere with an iridescent surface. ... In mathematics, Plateaus problem is to show the existence of a minimal surface with a given boundary. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... The Weaire-Phelan structure is a complex 3-dimensional structure discovered in 1993 by Denis Weaire and Robert Phelan, 2 physicists based at Trinity College Dublin, using computer simulations of foam. ... Tensile architecture is a relatively new field of architecture devoted to lightweight membrane structures. ... In mathematics, the Enneper-Weierstrass parameterization of minimal surfaces is a classical piece of differential geometry. ...

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. Retrieved on December 27, 2006. (graphical introduction to minimal surfaces and soap films.)
• Various (2000-). EG-Models. Retrieved on 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. Retrieved on April 15, 2006. (Describes the discovery of Costa's surface)
• Martin Steffens and Christian Teitzel. Grape Minimal Surface Library. Retrieved on April 15, 2006. (An collection of minimal surfaces)
• David Hoffman, Jim Hoffman et. al.. Scientific Graphics Project. Retrieved on April 24, 2006. (An collection of minimal surfaces with classical and modern examples)
• Jacek Klinowski. Periodic Minimal Surfaces Gallery. Retrieved on December 15, 2006. (An collection of minimal surfaces with classical and modern examples)