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.
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:
“How easily it falls, how easily I let drift
On the surface of morning feathers of self-reproach:
How easily I disperse the scolding of snow.”
—Philip Larkin (1922–1986)
“See how peaceful it is here. The sea is everything. An immense reservoir of nature where I roam at will.... Think of it. On the surface there is hunger and fear. Men still exercise unjust laws. They fight, tear one another to pieces. A mere few feet beneath the waves their reign ceases, their evil drowns. Here on the ocean floor is the only independence. Here I am free.”
—Earl Felton, and Richard Fleischer. Captain Nemo (James Mason)
“Brave men are all vertebrates; they have their softness on the surface and their toughness in the middle.”
—Gilbert Keith Chesterton (1874–1936)