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:

“We’ve forgotten what it’s like not to be able to reach the light switch. We’ve forgotten a lot of the monsters that seemed to live in our room at night. Nevertheless, those memories are still there, somewhere inside us, and can sometimes be brought to the *surface* by events, sights, sounds, or smells. Children, though, can never have grown-up feelings until they’ve been allowed to do the growing.”

—Fred Rogers (20th century)

“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.

“Bees

Shaking the heavy dews from bloom and frond.

Boys

Bursting the *surface* of the ebony pond.”

—Wilfred Owen (1893–1918)