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

### Famous quotes containing the word surface:

“Here Men from The Planet Earth

First Set Foot upon The Moon

July, 1969 AD

We Came in Peace for All Mankind”

—Plaque left behind on the moon’s *surface* by the crew of Apollo 11.

“Just under the *surface* I shall be, all together at first, then separate and drift, through all the earth and perhaps in the end through a cliff into the sea, something of me.”

—Samuel Beckett (1906–1989)

“In Manhattan, every flat *surface* is a potential stage and every inattentive waiter an unemployed, possibly unemployable, actor.”

—Quentin Crisp (b. 1908)