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 R3 — 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.
Famous quotes containing the word surface:
“Brave men are all vertebrates; they have their softness on the surface and their toughness in the middle.”
—Gilbert Keith Chesterton (18741936)
“These wonderful things
Were planted on the surface of a round mind that was to become our present time.”
—John Ashbery (b. 1927)
“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 (18171862)