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**

... The field's

**surface**, originally composed of AstroTurf, contained many gaps and uneven patches ... 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 ...

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

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

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

**surface**of the disk ...

**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 ... (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 ... point, and each of the xy-, yz-, and xz-planes are tangential to the

**surface**there ...

... In surveying and geodesy, a datum is a set of reference points on the Earth's

**surface**against which position measurements are made and (often) an ... 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**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 ...

### Famous quotes containing the word surface:

“And yet we constantly reclaim some part of that primal spontaneity through the youngest among us, not only through their sorrow and anger but simply through everyday discoveries, life unwrapped. To see a child touch the piano keys for the first time, to watch a small body slice through the *surface* of the water in a clean dive, is to experience the shock, not of the new, but of the familiar revisited as though it were strange and wonderful.”

—Anna Quindlen (b. 1952)

“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)

“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)