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:
“A lifeless planet. And yet, yet still serving a useful purpose, I hope. Yes, a sun. Warming the surface of some other world. Giving light to those who may need it.”
—Franklin Coen, and Joseph Newman. Exeter (Jeff Morrow)
“Nature centres into balls,
And her proud ephemerals,
Fast to surface and outside,
Scan the profile of the sphere;
Knew they what that signified,
A new genesis were here.”
—Ralph Waldo Emerson (1803–1882)
“Puritanism, in whatever expression, is a poisonous germ. On the surface everything may look strong and vigorous; yet the poison works its way persistently, until the entire fabric is doomed.”
—Emma Goldman (1869–1940)