In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
(see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a
soap film
Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may
self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see
minimal surface of revolution
In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in t ...
): the standard definitions only relate to a
local optimum
In applied mathematics and computer science, a local optimum of an optimization problem is a solution that is optimal (either maximal or minimal) within a neighboring set of candidate solutions. This is in contrast to a global optimum, which ...
, not a
global optimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
.
Definitions
Minimal surfaces can be defined in several equivalent ways in R
3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
calculus of variations,
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
.
:Local least area definition: A surface ''M'' ⊂ R
3 is minimal if and only if every point ''p'' ∈ ''M'' has a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
, bounded by a simple closed curve, which has the least area among all surfaces having the same boundary.
This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area.
:Variational definition: A surface ''M'' ⊂ R
3 is minimal if and only if it is a
critical point of the area
functional for all compactly supported
variations
Variation or Variations may refer to:
Science and mathematics
* Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon
* Genetic variation, the difference in DNA among individua ...
.
This definition makes minimal surfaces a 2-dimensional analogue to
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, which are analogously defined as critical points of the length functional.
:Mean curvature definition: A surface ''M'' ⊂ R
3 is minimal if and only if its
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
is equal to zero at all points.
A direct implication of this definition is that every point on the surface is a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
with equal and opposite
principal curvatures. Additionally, this makes minimal surfaces into the static solutions of
mean curvature flow. By the
Young–Laplace equation
In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or w ...
, the
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
of a soap film is proportional to the difference in pressure between the sides. If the soap film does not enclose a region, then this will make its mean curvature zero. By contrast, a spherical
soap bubble encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature.
:Differential equation definition: A surface ''M'' ⊂ R
3 is minimal if and only if it can be locally expressed as the graph of a solution of
::
The partial differential equation in this definition was originally found in 1762 by
Lagrange,
[J. L. Lagrange. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. Miscellanea Taurinensia 2, 325(1):173{199, 1760.] and
Jean Baptiste Meusnier
Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...
discovered in 1776 that it implied a vanishing mean curvature.
[J. B. Meusnier. Mémoire sur la courbure des surfaces. Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans, 10:477–510, 1785. Presented in 1776.]
:Energy definition: A
conformal immersion ''X'': ''M'' → R
3 is minimal if and only if it is a critical point of the
Dirichlet energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
for all compactly supported variations, or equivalently if any point ''p'' ∈ ''M'' has a neighbourhood with least energy relative to its boundary.
This definition ties minimal surfaces to
harmonic functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \fr ...
and
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
.
:Harmonic definition: If ''X'' = (''x''
1, ''x''
2, ''x''
3): ''M'' → R
3 is an
isometric immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
of a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
into 3-space, then ''X'' is said to be minimal whenever ''x
i'' is a
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
on ''M'' for each ''i''.
A direct implication of this definition and the
maximum principle for harmonic functions is that there are no
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
minimal surfaces in R
3.
:Gauss map definition: A surface ''M'' ⊂ R
3 is minimal if and only if its
stereographically projected
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...
''g'': ''M'' → C ∪ {∞} is
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
with respect to the underlying
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
structure, and ''M'' is not a piece of a sphere.
This definition uses that the mean curvature is half of the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
, which is linked to the derivatives of the Gauss map. If the projected Gauss map obeys the
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
then either the trace vanishes or every point of ''M'' is
umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
, in which case it is a piece of a sphere.
The local least area and variational definitions allow extending minimal surfaces to other
Riemannian manifolds
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
than R
3.
History
Minimal surface theory originates with
Lagrange who in 1762 considered the variational problem of finding the surface ''z'' = ''z''(''x'', ''y'') of least area stretched across a given closed contour. He derived the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
for the solution
:
He did not succeed in finding any solution beyond the plane. In 1776
Jean Baptiste Marie Meusnier
Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...
discovered that the
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarit ...
and
catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
satisfy the equation and that the differential expression corresponds to twice the
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
of the surface, concluding that surfaces with zero mean curvature are area-minimizing.
By expanding Lagrange's equation to
:
Gaspard Monge
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...
and
Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by
Heinrich Scherk in 1830 to derive his
surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
* Sur ...
, they were generally regarded as practically unusable.
Catalan
Catalan may refer to:
Catalonia
From, or related to Catalonia:
* Catalan language, a Romance language
* Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia
Places
* 13178 Catalan, asteroid #1 ...
proved in 1842/43 that the helicoid is the only
ruled minimal surface.
Progress had been fairly slow until the middle of the century when the
Björling problem was solved using complex methods. The "first golden age" of minimal surfaces began.
Schwarz found the solution of the
Plateau problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is ...
for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 (allowing the construction of his periodic
surface families) using complex methods.
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
and
Enneper developed more useful
representation formulas, firmly linking minimal surfaces to
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
harmonic functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \fr ...
. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten.
Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by
Jesse Douglas
Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem.
Life and career
He was born to a Jewish family in New York City, the son of Sarah (née ...
and
Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.
Biography
Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
was a major milestone.
Bernstein's problem
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimens ...
and
Robert Osserman
Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces.
Raised in Bronx, he went to Bronx High School of ...
's work on complete minimal surfaces of finite total curvature were also important.
Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of
a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R
3 of finite topological type. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the
conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). Another cause was the verification by H. Karcher that the
triply periodic minimal surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations.
These surfaces have the symmetries of a crystallographic group. Numerous examples are know ...
s originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them.
Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. the
positive mass conjecture
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...
, the
Penrose conjecture) and three-manifold geometry (e.g. the
Smith conjecture
In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), knot.
showed that a non-trivial or ...
, the
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured ...
, the
Thurston Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...
).
Examples
Classical examples of minimal surfaces include:
* the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, which is a
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
case
*
catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
s: minimal surfaces made by rotating a
catenary
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superfici ...
once around its directrix
*
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarit ...
s: 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
Surfaces from the 19th century golden age include:
*
Schwarz minimal surface In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named ...
s:
triply periodic surfaces that fill R
3
*
Riemann's minimal surface: A posthumously described periodic surface
* the
Enneper surface
In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by:
\begin
x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\
y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...
* the
Henneberg surface: the first non-orientable minimal surface
*
Bour's minimal surface
* the
Neovius surface: a triply periodic surface
Modern surfaces include:
* the
Gyroid
A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.
History and properties
The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces. I ...
: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for liquid crystal structure
* the
Saddle tower family: generalisations of
Scherk's second surface
*
Costa's minimal surface
In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact s ...
: Famous conjecture disproof. Described in 1982 by
Celso Costa
Celso José da Costa (born April 7, 1949 in Congonhinhas) is a Brazilian mathematician working in differential geometry. His research activity has focused in the construction and classification of minimal surfaces embedded in three-dimensional Eu ...
and later visualized by
Jim Hoffman. Jim Hoffman, David Hoffman and William Meeks III then extended the definition to produce a family of surfaces with different rotational symmetries.
* the
Chen–Gackstatter surface family, adding handles to the Enneper surface.
Generalisations and links to other fields
Minimal surfaces can be defined in other
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
than R
3, such as
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
, higher-dimensional spaces or
Riemannian manifolds
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
.
The definition of minimal surfaces can be generalized/extended to cover
constant-mean-curvature surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics ...
s: surfaces with a constant mean curvature, which need not equal zero.
The curvature lines of an isothermal surface form an isothermal net.
In
discrete differential geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, g ...
discrete minimal surfaces are studied:
simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions. Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known.
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces.
Minimal surfaces have become an area of intense scientific study, especially in the areas of
molecular engineering
Molecular engineering is an emerging field of study concerned with the design and testing of molecular properties, behavior and interactions in order to assemble better materials, systems, and processes for specific functions. This approach, in whi ...
and
materials science, due to their anticipated applications in
self-assembly
Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...
of complex materials. The
endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.
In the fields of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and
Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as
apparent horizon
In general relativity, an apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards and those directed outward but moving inward.
Apparent horizons are not invariant properties of spacetim ...
s, are significant. In contrast to the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
, they represent a
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
-based approach to understanding
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
boundaries.
Structures with minimal surfaces can be used as tents.
Minimal surfaces are part of the
generative design
Generative design is an iterative design process that involves a program that will generate a certain number of outputs that meet certain constraints, and a designer that will fine tune the feasible region by selecting specific output or chang ...
toolbox used by modern designers. In architecture there has been much interest in
tensile structure
A tensile structure is a construction of elements carrying only tension and no compression or bending. The term ''tensile'' should not be confused with tensegrity, which is a structural form with both tension and compression elements. Tensile ...
s, which are closely related to minimal surfaces. Notable examples can be seen in the work of
Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
,
Shigeru Ban
[Biography](_blank)
, The Hyatt Foundation, retrieved 26 March 2014 is a Japanese architect, known for his i ...
, and
Zaha Hadid
Dame Zaha Mohammad Hadid ( ar, زها حديد ''Zahā Ḥadīd''; 31 October 1950 – 31 March 2016) was an Iraqi-British architect, artist and designer, recognised as a major figure in architecture of the late 20th and early 21st centu ...
. The design of the
Munich Olympic Stadium
Olympiastadion () is a stadium located in Munich, Germany. Situated at the heart of the '' Olympiapark München'' in northern Munich, the stadium was the main venue for the 1972 Summer Olympics.
The original capacity was maximally and officiall ...
by Frei Otto was inspired by soap surfaces. Another notable example, also by Frei Otto, is the German Pavilion at
Expo 67
The 1967 International and Universal Exposition, commonly known as Expo 67, was a general exhibition from April 27 to October 29, 1967. It was a category One World's Fair held in Montreal, Quebec, Canada. It is considered to be one of the most su ...
in Montreal, Canada.
In the art world, minimal surfaces have been extensively explored in the sculpture of
(1927–2018),
Robert Longhurst (1949– ), and
Charles O. Perry (1929–2011), among others.
See also
*
Bernstein's problem
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimens ...
*
Bilinear interpolation
In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., ''x'' and ''y'') using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be ...
*
Bryant surface
*
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
*
Enneper–Weierstrass parameterization
*
Harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
*
Harmonic morphism In mathematics, a harmonic morphism is a (smooth) map \phi:(M^m,g)\to (N^n,h) between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class o ...
*
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...
*
Schwarz minimal surface In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named ...
*
Soap bubble
*
Surface Evolver
Surface Evolver is an interactive program for the study of Surface (topology), surfaces shaped by surface tension and other energies, and subject to various constraints. A surface is implemented as a simplicial complex. The user defines an initial ...
*
Stretched grid method
*
Tensile structure
A tensile structure is a construction of elements carrying only tension and no compression or bending. The term ''tensile'' should not be confused with tensegrity, which is a structural form with both tension and compression elements. Tensile ...
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Triply periodic minimal surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations.
These surfaces have the symmetries of a crystallographic group. Numerous examples are know ...
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Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
References
Further reading
Textbooks
* Tobias Holck Colding and William P. Minicozzi, II. ''A course in minimal surfaces.'' Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp.
* R. Courant. ''Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces.'' Appendix by M. Schiffer. Interscience Publishers, Inc., New York, N.Y., 1950. xiii+330 pp.
* Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny. ''Minimal surfaces.'' Revised and enlarged second edition. With assistance and contributions by A. Küster and R. Jakob. Grundlehren der Mathematischen Wissenschaften, 339. Springer, Heidelberg, 2010. xvi+688 pp. , ,
* H. Blaine Lawson, Jr. ''Lectures on minimal submanifolds. Vol. I.'' Second edition. Mathematics Lecture Series, 9. Publish or Perish, Inc., Wilmington, Del., 1980. iv+178 pp.
* Johannes C.C. Nitsche. ''Lectures on minimal surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems.'' Translated from the German by Jerry M. Feinberg. With a German foreword. Cambridge University Press, Cambridge, 1989. xxvi+563 pp.
*Robert Osserman. ''A survey of minimal surfaces.'' Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp. ,
Online resources
* ''(graphical introduction to minimal surfaces and soap films.)''
* ''(A collection of minimal surfaces with classical and modern examples)''
* ''(A collection of minimal surfaces)''
* ''(Online journal with several published models of minimal surfaces)''
External links
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3D-XplorMath-J Homepage — Java program and applets for interactive mathematical visualisation
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Differential geometry
Differential geometry of surfaces