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:

“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)

“We say justly that the weak person is flat, for, like all flat substances, he does not stand in the direction of his strength, that is, on his edge, but affords a convenient *surface* to put upon. He slides all the way through life.... But the brave man is a perfect sphere, which cannot fall on its flat side and is equally strong every way.”

—Henry David Thoreau (1817–1862)

“If the man who paints only the tree, or flower, or other *surface* he sees before him were an artist, the king of artists would be the photographer. It is for the artist to do something beyond this: in portrait painting to put on canvas something more than the face the model wears for that one day; to paint the man, in short, as well as his features.”

—James Mcneill Whistler (1834–1903)