## ::Surface

### ::concepts

**Surface**::surfaces Which::closed Sphere::space Boundary::plane Torus::compact '''P'''::complex

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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}, such as a sphere. 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.

**Surface sections**

Intro Definitions and first examples Extrinsically defined surfaces and embeddings Construction from polygons Connected sums Closed surfaces Surfaces in geometry See also Notes References External links

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