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

... 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 ... Two smooth

**surfaces**are diffeomorphic if and only if they are homeomorphic ...

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

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

**surface**of the slider ... 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 ...

... 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 ... Horizontal datums are used for describing a point on the Earth's

**surface**, in latitude and longitude or another coordinate system ... 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 ... 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 ... 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)

“The *surface* of the earth is soft and impressible by the feet of men; and so with the paths which the mind travels. How worn and dusty, then, must be the highways of the world, how deep the ruts of tradition and conformity!”

—Henry David Thoreau (1817–1862)

“There’s something tragic in the fate of almost every person—it’s just that the tragic is often concealed from a person by the banal *surface* of life.... A woman will complain of indigestion and not even know that what she means is that her whole life has been shattered.”

—Ivan Sergeevich Turgenev (1818–1883)