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":

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

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

**surface**of the slider ... of the air bearing is to maintain the flying height constant as the head moves over the

**surface**of the disk ...

**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**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 ...

... and geodesy, a datum 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 ... Horizontal 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 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-dimensiona ... 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 ...

### Famous quotes containing the word surface:

“But the *surface* of the Earth was meant for man. He wasn’t meant to live in a hole in the ground.”

—Edward L. Bernds (b. 1911)

“If the man who paints only the tree, or flower, or other *surface* he sees before him were an artist, the king of artists would be the photographer. It is for the artist to do something beyond this: in portrait painting to put on canvas something more than the face the model wears for that one day; to paint the man, in short, as well as his features.”

—James Mcneill Whistler (1834–1903)

“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)