# Dictionary Definition

surface adj

1 on the surface; "surface materials of the moon"
[ant: subsurface,
overhead]

2 involving a surface only; "her beauty is only
skin-deep"; "superficial bruising"; "a surface wound" [syn:
skin-deep,
superficial,
surface(a)]

### Noun

1 the outer boundary of an artifact or a material
layer constituting or resembling such a boundary; "there is a
special cleaner for these surfaces"; "the cloth had a pattern of
red dots on a white surface"

2 the extended two-dimensional outer boundary of
a three-dimensional object; "they skimmed over the surface of the
water"; "a brush small enough to clean every dental surface"; "the
sun has no distinct surface"

3 the outermost level of the land or sea;
"earthquakes originate far below the surface"; "three quarters of
the Earth's surface is covered by water" [syn: Earth's
surface]

4 a superficial aspect as opposed to the real
nature of something; "it was not what it appeared to be on the
surface"

5 information that has become public; "all the
reports were out in the open"; "the facts had been brought to the
surface" [syn: open]

6 a device that provides reactive force when in
motion relative to the surrounding air; can lift or control a plane
in flight [syn: airfoil,
aerofoil, control
surface]

### Verb

2 put a coat on; cover the surface of; furnish
with a surface; "coat the cake with chocolate" [syn: coat]

3 appear or become visible; make a showing; "She
turned up at the funeral"; "I hope the list key is going to surface
again" [syn: come on, come out,
turn up,
show
up]

# User Contributed Dictionary

## English

### Etymology

From surface.### Pronunciation

- /sɜːfəs̩/

### Noun

#### Translations

up-side of a flat object

- Arabic:
- Bosnian: površina
- Catalan: superfície
- Chinese: 表面 (biǎomiàn)
- Croatian: površina
- Czech: povrch
- Dutch: oppervlakte
- Finnish: pinta (1,2)
- French: surface
- German: Oberfläche
- Hungarian: felszín
- Ido: surfaco
- Italian: superficie
- Japanese: 表面 (ひょうめん, hyōmen)
- Korean: 표면 (pyomyoen)
- Malayalam: ഉപരിതലം (uparithalam)
- Norwegian: overflate
- Polish: powierzchnia
- Portuguese: superfície
- Russian: поверхность (povérχnost’)
- Serbian:
- Slovak: povrch
- Slovene: površina
- Spanish: superficie
- Swedish: yta
- Telugu: ఉపరితలం (uparitalam)
- Turkish: yüzey

### Verb

#### Translations

to rise to the surface

for information to become known

- Russian: всплывать (vsplyvát’), выясняться (vyjasnját’sja)

## French

### Etymology

### Pronunciation

### Noun

fr-noun f# Extensive Definition

In mathematics, specifically in
topology, a surface is
a two-dimensional
manifold. The most
familiar examples are those that arise as the boundaries of solid
objects in ordinary three-dimensional Euclidean
space, E³. On the other hand, there are also more exotic
surfaces, that are so "contorted" that they cannot be embedded in three-dimensional
space at all.

To say that a surface is "two-dimensional" means
that, about each point, there is a coordinate patch on which a
two-dimensional coordinate
system is defined. For example, the surface of the Earth is (ideally) a
two-dimensional sphere,
and latitude and
longitude provide
coordinates on it — except at the International
Date Line and the poles, where longitude is undefined. This
example illustrates that in general it is not possible to extend
any one coordinate patch to the entire surface; surfaces, like
manifolds of all dimensions, are usually constructed by patching
together multiple coordinate systems.

Surfaces find application in physics, engineering, computer
graphics, and many other disciplines, primarily when they
represent the surfaces of physical objects. For example, in
analyzing the aerodynamic properties of
an airplane, the
central consideration is the flow of air along its surface.

## Definitions and first examples

A (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closed half space of E² (Euclidean 2-space). The neighborhood, along with the homeomorphism to Euclidean space, is called a (coordinate) chart.The set of points that have an open neighbourhood
homeomorphic to E² is called the interior of the surface; it is
always non-empty. The
complement
of the interior is called the boundary; it is a one-manifold, or
union of closed curves. The simplest example of a surface with
boundary is the closed disk
in E²; its boundary is a circle.

A surface with an empty boundary is called
boundaryless. (Sometimes the word surface, used alone, refers only
to boundaryless surfaces.) A closed surface is one that is
boundaryless and compact.
The two-dimensional sphere, the two-dimensional torus, and the real
projective plane are examples of closed surfaces.

The Möbius
strip is a surface with only one "side". In general, a surface
is said to be orientable if it does not contain a homeomorphic copy
of the Möbius strip; intuitively, it has two distinct "sides". For
example, the sphere and torus are orientable, while the real
projective plane is not (because deleting a point or disk from the
real projective plane produces the Möbius strip).

More generally, it is common in differential
and algebraic
geometry to study surfaces with
singularities, such as self-intersections, cusps, etc.

## Extrinsically defined surfaces and embeddings

Historically, surfaces were originally defined
and constructed not using the abstract, intrinsic definition given
above, but extrinsically, as subsets of Euclidean spaces such as
E³.

Let f be a continuous, injective
function from R² to R³. Then the image
of f is said to be a parametric
surface. A surface
of revolution can be viewed as a special kind of parametric
surface.

On the other hand, suppose that f is a smooth
function from R³ to R whose gradient is nowhere zero. Then
the locus
of zeros
of f is said to be an implicit
surface. If the condition of non-vanishing gradient is dropped
then the zero locus may develop singularities.

One can also define parametric and implicit
surfaces in higher-dimensional Euclidean spaces En. It is natural
to ask whether all surfaces (defined abstractly, as in the
preceding section) arise as subsets of some En. The answer is yes;
the Whitney
embedding theorem, in the case of surfaces, states that any
surface can be embedded homeomorphically into E4. Therefore the
extrinsic and intrinsic approaches turn out to be equivalent.

In fact, any compact surface that is either
orientable or has a boundary can be embedded in E³; on the other
hand, the real projective plane, which is compact, non-orientable
and without boundary, cannot be embedded into E³ (see Gramain).
Steiner
surfaces, including Boy's
surface, the Roman
surface and the cross-cap, are
immersions of the real
projective plane into E³. These surfaces are singular where the
immersions intersect themselves.

The Alexander
horned sphere is a well-known pathological
embedding of the two-sphere into the three-sphere.

The chosen embedding (if any) of a surface into
another space is regarded as extrinsic information; it is not
essential to the surface itself. For example, a torus can be
embedded into E³ in the "standard" manner (that looks like a
bagel) or in a knotted
manner (see figure). The two embedded tori are homeomorphic but not
isotopic; they are
topologically equivalent, but their embeddings are not.

## Construction from polygons

Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (A with A, B with B), so that the arrows point in the same direction, yields the indicated surface.Any fundamental polygon can be written
symbolically as follows. Begin at any vertex, and proceed around
the perimeter of the polygon in either direction until returning to
the starting vertex. During this traversal, record the label on
each edge in order, with an exponent of -1 if the edge points
opposite to the direction of traversal. The four models above, when
traversed clockwise starting at the upper left, yield

- sphere: A B B^ A^
- real projective plane: A B A B
- torus: A B A^ B^
- Klein bottle: A B A B^.

The expression thus derived from a fundamental
polygon of a surface turns out to be the sole relation in a
presentation of the fundamental
group of the surface with the polygon edge labels as
generators. This is a consequence of the
Seifert–van Kampen theorem.

## Quotients and connected summation

Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum.The connected
sum of two surfaces M and N, denoted M # N, is obtained by
removing a disk from each of them and gluing them along the
boundary components that result. The Euler
characteristic \chi of M # N is the sum of the Euler
characteristics of the summands, minus two:

- \chi(M \# N) = \chi(M) + \chi(N) - 2.\,

The sphere S is an identity
element for the connected sum, meaning that S # M = M. This is
because deleting a disk from the sphere leaves a disk, which simply
replaces the disk deleted from M upon gluing.

Connected summation with the torus T has the
effect of attaching a "handle" to the other summand M. If M is
orientable, then so is T # M. The connected sum can be iterated to
attach any number g of handles to M.

The connected sum of two real projective planes
is the Klein bottle. The connected sum of the real projective plane
and the Klein bottle is homeomorphic to the connected sum of the
real projective plane with the torus. Any connected sum involving a
real projective plane is nonorientable.

## Classification of closed surfaces

The classification of closed surfaces states that any closed surface is homeomorphic to some member of one of these three families:- the sphere;
- the connected sum of g tori, for g \geq 1;
- the connected sum of k real projective planes, for k \geq 1.

The surfaces in the first two families are
orientable. It is convenient to combine the two families by
regarding the sphere as the connected sum of 0 tori. The number g
of tori involved is called the genus of the surface. Since the
sphere and the torus have Euler characteristics 2 and 0,
respectively, it follows that the Euler characteristic of the
connected sum of g tori is
2 − 2g.

The surfaces in the third family are
nonorientable. Since the Euler characteristic of the real
projective plane is 1, the Euler characteristic of the connected
sum of k of them is 2 − k.

It follows that a closed surface is determined,
up to homeomorphism, by two pieces of information: its Euler
characteristic, and whether it is orientable or not. In other
words, Euler characteristic and orientability completely classify
closed surfaces up to homeomorphism.

## Surfaces in geometry

Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E². This elaboration allows calculus to be applied to surfaces to prove many results.Two smooth surfaces are diffeomorphic if and only
if they are homeomorphic. (The analogous result does not hold for
higher-dimensional manifolds.) Thus closed
surfaces are classified up to diffeomorphism by their Euler
characteristic and orientability.

Smooth surfaces equipped with Riemannian
metrics are of foundational importance in differential
geometry. A Riemannian metric endows a surface with notions of
geodesic, distance, angle, and area. It also gives
rise to Gaussian
curvature, which describes how curved or bent the surface is at
each point. Curvature is a rigid, geometric property, in that it is
not preserved by general diffeomorphisms of the surface. However,
the famous Gauss-Bonnet
theorem for closed surfaces states that the integral of the
Gaussian curvature K over the entire surface S is determined by the
Euler characteristic:

- \int_S K \; dA = 2 \pi \chi(S).

Another way in which surfaces arise in geometry
is by passing into the complex domain. A complex one-manifold is a
smooth oriented surface, also called a Riemann
surface. Any complex nonsingular algebraic
curve viewed as a real manifold is a Riemann surface.

Every closed surface admits complex structures.
Complex structures on a closed oriented surface correspond to
conformal
equivalence classes of Riemannian metrics on the surface. One
version of the uniformization
theorem (due to Poincaré)
states that any Riemannian
metric on an oriented, closed surface is conformally equivalent
to an essentially unique metric of constant
curvature. This provides a starting point for one of the
approaches to Teichmüller
theory, which provides a finer classification of Riemann
surfaces than the topological one by Euler characteristic
alone.

A complex surface is a complex two-manifold and
thus a real four-manifold; it is not a surface in the sense of this
article. Neither are algebraic curves or surfaces defined over
fields
other than the complex numbers.

## See also

- Volume form, for volumes of surfaces in En
- Poincaré metric, for metric properties of Riemann surfaces
- Area element, the area of a differential element of a surface

## References

- Topology of Surfaces (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")
- Topology and Geometry
- A Basic Course in Algebraic Topology

## External links

- Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
- 3D Graph Explorer, a free java applet/application to explore mathematically defined surfaces

surface in Asturian: Superficie

surface in Catalan: Superfície

surface in Czech: Povrch

surface in German: Fläche (Topologie)

surface in Spanish: Superficie

surface in French: Surface

surface in Friulian: Superficie

surface in Galician: Superficie

surface in Ido: Surfaco

surface in Interlingua (International Auxiliary
Language Association): Superficie

surface in Italian: Superficie
(matematica)

surface in Latvian: Virsma

surface in Japanese: 表面

surface in Polish: Powierzchnia

surface in Portuguese: Superfície

surface in Romanian: Suprafaţă

surface in Russian: Поверхность

surface in Simple English: Surface

surface in Slovak: Povrch

surface in Slovenian: Ploskev

surface in Serbian: Површ

surface in Vietnamese: Mặt

surface in Ukrainian: Поверхня

surface in Venetian: Superficie

surface in Chinese: 曲面

# Synonyms, Antonyms and Related Words

3-D, acreage, ankle-deep, apparent, appear, appearing, area, arise, bail out, bob up, border, boundary, breadth, break cover, break
forth, break water, burst forth, circumference, clerestory, come, come forth, come out, come
up, concrete, continuum, conventionalized,
cortex, cortical, cover, covering, crop up, crust, cubic, cursoriness, cursory, debouch, depthless, dimension, dimensional, disembogue, dive, effuse, emanate, emerge, emptiness, empty space,
envelope, epidermal, epidermic, epidermis, erupt, exomorphic, expanse, expansion, extension, extent, exterior, exteriority, external, extrinsic, extrinsically, extrude, facade, face, facet, feel, field, finish, flat, float up, flood negative,
flood the tanks, fly up, formal, formalist, formalistic, formulary, fountain, fourth-dimensional,
fringe, front, galactic space, gloss, go below, grain, granular texture, gush, impersonal, indentation, infinite space,
integument, interface, interstellar space,
issue, issue forth,
jejune, jet, jump out, jump up, knee-deep,
knub, leap up, legalistic, light, lineaments, materialize, measure, mere scratch, nap, no depth, no water, nominal, not deep, nothingness, nub, on the surface, open, ostensible, ostensibly, out, outer, outer face, outer layer,
outer side, outer skin, outer space, outermost, outline, outlying, outmost, outside, outstanding, outward, outward-facing,
outwardly, pave, pedantic, peripheral, periphery, pile, pinprick, pit, plane, pock, pop up, proportion, proportional, protrude, protuberance, public, ridgepole, rig for diving,
rind, rise, rocket, roof, roofpole, rooftop, roundabout, sally, sally forth, scratch, seeming, shag, shallow, shallow-rooted,
shallowness,
shell, shoal, shoaliness, shoot up, show
up, side, skin, skin-deep, skyrocket, slight, slightness, space, space-time, spatial, spatial extension,
spatiotemporal,
sphere, spherical, spread, spring up, spurt, start up, stereoscopic, structure, stylized, submerge, superficial, superficial
extension, superficiality, superficially, superficies, superstratum, surface
texture, tarmac, texture, thin, three-dimensional, to all
appearances, top, top floor,
top side, topside,
topsides, tract, trivial, triviality, two-dimensional,
unprofound, upleap, upper side, upshoot, upside, upspear, upspring, upstart, vault up, veneer, visible, void, volume, volumetric, wale, weave, woof