In mathematics, specifically in topology, a **surface** is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space **R**3 — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.

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 two-dimensional coordinates on it (except at the poles and along the 180th meridian).

The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing 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.

Read more about Surface: Definitions and First Examples, Extrinsically Defined Surfaces and Embeddings, Construction From Polygons, Connected Sums, Closed Surfaces, Surfaces in Geometry

### Other articles related to "surface, surfaces":

**Surface**s 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 ...

... is a set of reference points on the Earth's

**surface**against which position measurements are made and (often) an associated model of the shape of the Earth (reference ellipsoid) to define a geodetic ... datums are used for describing a point on the Earth's

**surface**, in latitude and longitude or another coordinate system ... In engineering and drafting, a datum is a reference point,

**surface**, or axis on an object against which measurements are made ...

**Surface**

... The field's

**surface**, originally composed of AstroTurf, contained many gaps and uneven patches ... Baseball players also complained about the

**surface**... It was much harder than other AstroTurf

**surfaces**, and the shock of running on it often caused back pain ...

**Surface**

... The Roman

**surface**or Steiner

**surface**(so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an ... longitude (θ) and latitude (φ), gives parametric equations for the Roman

**surface**as follows x = r2 cos θ cos φ sin φ y = r2 sin θ cos φ sin φ z = r2 cos θ sin θ cos2 φ ... The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the

**surface**there ...

... In a hard drive, the heads 'fly' above the disk

**surface**with clearance of as little as 3 nanometres ... by the design of an air-bearing etched onto the disk-facing

**surface**of the slider ... constant as the head moves over the

**surface**of the disk ...

### Famous quotes containing the word surface:

“Bees

Shaking the heavy dews from bloom and frond.

Boys

Bursting the *surface* of the ebony pond.”

—Wilfred Owen (1893–1918)

“Voluptuaries, consumed by their senses, always begin by flinging themselves with a great display of frenzy into an abyss. But they survive, they come to the *surface* again. And they develop a routine of the abyss: “It’s four o’clock ... At five I have my abyss.””

—Colette [Sidonie Gabrielle Colette] (1873–1954)

“I cannot but conclude the bulk of your natives to be the most pernicious race of little, odious vermin that Nature ever suffered to crawl upon the *surface* of the earth.”

—Jonathan Swift (1667–1745)