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Topic: Minimal surface

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	Minimal surface - Wikipedia, the free encyclopedia
		In mathematics, a minimal surface is a surface with a mean curvature of zero.
		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.
		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.
	en.wikipedia.org /wiki/Minimal_surface (215 words)

	Triply Periodic Minimal Surfaces (Site not responding. Last check: 2007-11-07)
		A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece.
		The surfaces are of genus 3, 9, 15, 21, 27, and 33.
		The surfaces are of genus 6, 12, 18, 24, and 30.
	www.susqu.edu /facstaff/b/brakke/evolver/examples/periodic/periodic.html (1166 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		This notion can be made more precise by saying that the surface satisfies a set of differential equations, but another way of describing minimal surfaces is that their mean curvatures at every point is zero.
		Minimal surfaces are sometimes called "soap-bubble films," in the sense that these surfaces minimize their surface area for a given boundary.
		If you imagine a loop of wire shaped into a circle, then the minimal surface that spans the wire is a disk, which is a subset of the plane, which is a minimal surface.
	mathforum.org /library/drmath/view/51825.html (520 words)

	minimal surface
		A minimal surface has a mean curvature of zero.
		The first non-trivial examples of minimal surfaces, the catenoid and the helicoid, were discovered by the French geometer and engineer Jean Meusnier (1754-1793) in 1776, but there was then a gap of almost 60 years before the German Heinrich Scherk found some more.
		Most minimal surfaces are extremely hard to construct and visualize, in part because the majority of them are self-intersecting.
	www.daviddarling.info /encyclopedia/M/minimal_surface.html (314 words)

	Bour's Minimal Surface -- The 4th Dimension (Site not responding. Last check: 2007-11-07)
		It is a 1-parameter family minimal surfaces, associated with Bour's minimal curve.
		Like a minimal surface, a minimal curve is defined as having a mean curvature of zero.
		Bour's surface is closely related to Enneper's surface and Richmond's surface, as the associated families of both of these surfaces have special cases when they are identical to Bour's surface.
	www.coolphysics.com /4d/minimal/bour_surface/bour_surface.htm (326 words)

	Minimal Surfaces (Site not responding. Last check: 2007-11-07)
		Unbordered minimal surfaces have the property that each point is the center of a small patch that behaves like a soap-film relative to its boundary contour.
		From the point of view of local geometry, a minimal surface is equivalently described as one that is equally bent in all directions so as to have zero average curvature, e.g., a saddle shape.
		In Figure 11, we show new surfaces recently given by Hoffman, Wei, and Karcher [7]; the one-hole surface is the first complete, properly embedded minimal surface of finite topology and infinite total curvature to be found since the discovery of the helicoid in the eighteenth century.
	www.geom.uiuc.edu /docs/research/ieee94/node13.html (377 words)

	Newswise
		Minimal surfaces are proving to be important at the molecular level.
		Minimal surfaces are extremely stable as physical objects, Weber pointed out, and this can be an advantage in many kinds of structures.
		He has heard from architects who have seen computerized illustrations of some of his minimal surfaces and are intrigued by the possibility of adapting them to structures, both interior and exterior.
	www.newswise.com /articles/view/516913?sc=rssn (693 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		For example, that the normal to the surface extends continuously to the corner and is (at the corner) normal to the plane determined by the two polygonal sides.
		In the winter 1977-78, I continued to study Tromba's work on minimal surfaces, and began to work on the "finiteness problem", which is to prove (or disprove) the conjecture that no Jordan curve in three-space bounds an infinite number of relative minima of the area functional.
		Although many papers about minima minimal surfaces with polygonal boundaries have been written, there are still fundamental facts unknown--for example, we still don't have a good Morse theory of minimal surfaces with polygonal boundaries, despite various efforts in that direction over the past twenty years.
	www.mathcs.sjsu.edu /faculty/beeson/Papers/minsurf.html (2320 words)

	Costa's Minimal Surface -- The 4th Dimension (Site not responding. Last check: 2007-11-07)
		Costa's surface was an exceptionally rare discovery since almost all minimal surfaces have at least one if not all three of the properties of either being non-embedded, periodic, or constricted by boundaries.
		In fact, prior to the discovery of Costa's surface, the simple plane, helicoid, and catenoid were the were only three other known minimal surfaces that are unbounded, embedded, and non-periodic.
		Costa's surface is actually the most basic example in an infinitely large family of minimal surfaces.
	www.coolphysics.com /4d/minimal/costa_surface/costa_surface.htm (416 words)

	Level Set Methods: Minimal Surfaces and Soap Bubbles ------------------------------------------- Level Set Methods J.A. ... (Site not responding. Last check: 2007-11-07)
		The thin membrane that spans the wire boundary is a minimal surface; of all possible surfaces that span the boundary, it is the one with minimal energy.
		One way to think of this "minimal energy; is that to imagine the surface as an elastic rubber membrane: the minimal shape is the one that in which the rubber membrane is the most relaxed.
		An initial surface is attached to the wire frame, and then embedded as the zero level set of the signed distance function in all of space.
	math.berkeley.edu /~sethian/Movies/Movieminimal.html (486 words)

	Surface Story: Science News Online, Dec. 17, 2005 (Site not responding. Last check: 2007-11-07)
		This two-dimensional plane is the simplest example of a minimal surface that is infinite in extent and not an endless repetition of some basic shape.
		Also a minimal surface, it looks like a pair of intertwined spiral slides—a double helix—at its core and it stretches away to infinity as a stack of sheets.
		Several years ago, William H. Meeks III of the University of Massachusetts and Harold Rosenberg of the Université Denis Diderot in Paris proved that a complete, embedded minimal surface that was topologically a punctured sphere with no handles had to be either the basic helicoid or the plane.
	www.sciencenews.org /articles/20051217/bob9.asp (2050 words)

	Minimal Till It Melts
		Minimal surfaces with boundaries are a dime a dozen-almost any closed curve (in 3-space) surrounds something of smallest area-but unbounded minimal surfaces are harder to come by.
		The Costa surface turned out to be the first in an infinite family of new minimal surfaces.
		Like all minimal surfaces (except the plane), the Costa surface has negative curvature at every point: If the surface curves up in one direction, it curves down in the perpendicular direction, like the seat of a saddle.
	www.siam.org /siamnews/03-99/melts.htm (771 words)

	Catalan's Minimal Surface (Site not responding. Last check: 2007-11-07)
		Catalan's minimal surface is a minimal surface that contains a cycloid as a geodesic.
		, the tangent plane to the surface is parameterized by:
		The infinitesimal area of a patch on the surface is given by
	www.math.hmc.edu /faculty/gu/curves_and_surfaces/surfaces/catalan.html (36 words)

	Image Credits (Site not responding. Last check: 2007-11-07)
		The minimal surface can be continued analytically by reflection across any of the straight line segments on its boundary.
		Minimal surfaces, perfect mathematical representations for soap films, are not only of interest because of their significance in mathematics and their visual beauty but also because of their striking similarity to the real interfaces and separating membranes that are so abundant in nature and science.
		The subject of minimal surfaces is vast; many of the major mathematicians of the last three centuries have fallen under its spell.
	math.lib.umn.edu /imagcred.html (675 words)

	Soap Film
		There is a useful analogy to functions of two variables: A minimal surface corresponds to a local minimum of a function of two variables, and a surface of zero mean curvature corresponds to a critical point (a local minimum or a saddle point).
		Analogous to "every minimal surface has zero mean curvature" is the fact that every local minimum of a function of two variables is a critical point.
		My minimal surface is a local minimum of the area functional, whereas a surface with zero mean curvature is a critical point of the area functional.
	oak.ucc.nau.edu /jws8/dpgraph/soap_film.html (1023 words)

	Catalan's Minimal Surface -- The 4th Dimension (Site not responding. Last check: 2007-11-07)
		It is a 1-parameter family of isothermal minimal surfaces, associated with Catalan's minimal curve.
		Catalan's surface is an example of a periodic minimal surface that is immersed in 3-space
		Therefore, Catalan's surface is in a class of surfaces that have the property of being entirely composed of a single geodesic.
	www.coolphysics.com /4d/minimal/catalan_surface/catalan_surface.htm (236 words)

	SURFACE - Online Information article about SURFACE (Site not responding. Last check: 2007-11-07)
		peculiar to surfaces are scarcely of the like fundamental nature, being rather developments of the former set in their application to a more advanced portion of See also:
		In particular, there is nothing in plane geometry to correspond to the theory of the curves of curvature of a surface.
		Again, to the single theorem of plane geometry, that a line is the shortest distance between two points, there correspond in solid geometry two extensive and difficult theories—that of the geodesic lines on a surface and that of the minimal surface, or surface of minimum See also:
	encyclopedia.jrank.org /STE_SUS/SURFACE.html (545 words)

	Cartan's Corner : Spinors, Minimal Surfaces... and the Hopf Map. (Site not responding. Last check: 2007-11-07)
		The idea that minimal surfaces come in "conjugate" pairs, and the fact that these pairs represent extremes of "polarization" turns out to be a useful idea that generalizes the theory of coherent optics.
		The idea that a complex analytic function generates a minimal surface implies that all functional iterates are also minimal surfaces.
		The fact that the conjugate pairs of minimal surfaces are all related to the same light cone implies that the observable change of polarization state could proceed at speeds faster than the speed of light.
	www22.pair.com /csdc/car/carfre75.htm (370 words)

	Ivars Peterson's MathTrek - Minimal Snow
		The equations for this minimal surface were discovered in 1984 by the Brazilian mathematician Celso J. Costa.
		From certain angles, the Costa surface has the splendid elegance of a gracefully spinning dancer flinging out her full skirt so that it whirls parallel to the ground.
		And a minimal surface can also be thought of in terms of arches.
	www.maa.org /mathland/mathtrek_3_8_99.html (925 words)

	GANG \| Document Library: Series 3 Abstracts
		It has been shown in a series of recent papers [41, 31, 20, 24, 14, 7, 11] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations.
		A $K$-surface is a surface whose Gauss curvature $K$ is equal to a positive constant.
		The method of construction was generalized in [3] to obtain the first known examples of embedded, singly-periodic minimal surfaces with an infinite number of topological ends, invariant under screw-motions (with a nontrivial rotational component).
	www.gang.umass.edu /preprint/abstract3.html (2020 words)

	Math Trek: A Minimal Winter's Tale, Science News Online, Feb. 5, 2000 (Site not responding. Last check: 2007-11-07)
		A minimal surface is one whose area becomes greater whenever it is distorted.
		The particular minimal surface of interest to Wagon and his team had been discovered in 1864 by Alfred Enneper (1830—1885), a mathematics professor at the University of Göttingen in Germany.
		The equation defining the surface looks very simple, but the highly symmetric, complicated surface that results is hard to visualize because it curls around and intersects itself.
	www.sciencenews.org /20000205/mathtrek.asp (1003 words)

	Minimal Surfaces
		They have the property that their mean curvature is zero everywhere; and are thus a subset of CMC surfaces.
		Minimal Surface library, created using GRAPE at the University of Bonn.
		Triply Periodic Minimal Surfaces page at the Mathematics Department of Susquehanna University shows numerous examples of triply periodic minimal surfaces generated using the Brakke Surface Evolver.
	www.msri.org /about/sgp/jim/geom/minimal/main.html (192 words)

	Karsten Grosse-Brauckmann: Research (Site not responding. Last check: 2007-11-07)
		For many applications modelled with triply periodic surfaces (minimal, constant mean curvature, elastic, etc.) it is important to know about all surfaces with a given symmetry group, at least for low genus.
		For instance, minimal surfaces with genus 3+12k, having all the symmetries of the primitive lattice Z
		We also find compact cmc surfaces with the symmetry group of the Platonic polyhedra, such as tetrahedral symmetry (left) or dodecahedral symmetry (here is part of and the almost complete surface of genus 30).
	www.math.uni-bonn.de /people/kgb/Research/research.html (877 words)

	Mixing Materials and Mathematics
		The model surface with these properties is the helicoid-- the surface swept out by a horizontal line rotating at a constant rate as it moves at constant speed up a vertical axis.
		A geometer would condense this by defining a minimal surface as one that is ``locally area-minimizing.'' An engineer might think of a minimal surface as a membrane interface between two gases at the same pressure, which by the Laplace-Young law will have zero mean curvature (another way to characterize minimality).
		Among other things, by looking at level-set surfaces with zero replaced by a small value, they allow rough approximation of families of interface candididates whose mean curvature is expected to be close to constant, and which divide space into regions of unequal volume per unit cell.
	www.msri.org /about/sgp/david/papers/nature96 (1752 words)

	Getting a handle on minimal surfaces
		In a paper published in the Nov. 15 issue of Proceedings of the National Academy of Sciences, mathematicians Matthias Weber of Indiana University, David Hoffman of Stanford University, and Michael Wolf of Rice University presented a proof of the existence of a new minimal surface they call a genus one helicoid.
		All minimal surfaces have something important in common: a minimal surface area.
		"A minimal surface is formed when the pressure on both sides of a surface is the same," Weber explained.
	www.eurekalert.org /pub_releases/2005-12/iu-gah122005.php (655 words)

	Minimalist's Tug-of-War
		And the material boundary has the minimal surface area possible for a surface with the first two properties.
		To understand a minimal surface, think of a soap film on a wire frame bent into a complicated shape.
		The soap film is minimal because of surface tension, but the simulated minimal surface comes from the competition between the two properties and "is completely unexpected," says Torquato.
	focus.aps.org /story/v10/st26 (595 words)

	MIT OpenCourseWare \| Mathematics \| 18.994 Seminar in Geometry, Fall 2004 \| Projects
		He gives a lovely introduction to minimal surfaces, and then describes how the computer graphics pictures they were able to generate led to the discovery.
		This is not a theorem about minimal surfaces, but it is probably the most important theorem in surface theory, and it plays a role in projects 2 and 5 and is relevant to chapter 9 of Osserman, which is the last section we will cover in this course.
		Then minimal surfaces foliated by convex curves in parallel planes are considered, and Shiffman's beautiful theorems for this class of surfaces are presented.
	ocw.mit.edu /OcwWeb/Mathematics/18-994Fall-2004/Projects (1042 words)

	Formulation of the Minimal Surface Area Problem (Site not responding. Last check: 2007-11-07)
		The goal of this problem is to find a surface with minimal area for given boundary values.
		Since it is not always possible to calculate the exact minimum surface or write it as a simple function, we will compute an approximation of the surface.
		If you make the grid finer and finer, you will obtain a better approximation of the minimal surface unless the minimal surface is a plane.
	www-fp.mcs.anl.gov /otc/Guide/CaseStudies/msa/formulation.html (323 words)

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