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:
“Night City was like a deranged experiment in Social Darwinism, designed by a bored researcher who kept one thumb permanently on the fast-forward button. Stop hustling and you sank without a trace, but move a little too swiftly and you’d break the fragile surface tension of the black market; either way, you were gone ... though heart or lungs or kidneys might survive in the service of some stranger with New Yen for the clinic tanks.”
—William Gibson (b. 1948)
“These wonderful things
Were planted on the surface of a round mind that was to become our present time.”
—John Ashbery (b. 1927)
“White Pond and Walden are great crystals on the surface of the earth, Lakes of Light.... They are too pure to have a market value; they contain no muck. How much more beautiful than our lives, how much more transparent than our characters are they! We never learned meanness of them.”
—Henry David Thoreau (1817–1862)