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:
“It was a pretty game, played on the smooth surface of the pond, a man against a loon.”
—Henry David Thoreau (1817–1862)
“There’s something tragic in the fate of almost every person—it’s just that the tragic is often concealed from a person by the banal surface of life.... A woman will complain of indigestion and not even know that what she means is that her whole life has been shattered.”
—Ivan Sergeevich Turgenev (1818–1883)
“I cannot but conclude the bulk of your natives to be the most pernicious race of little, odious vermin that Nature ever suffered to crawl upon the surface of the earth.”
—Jonathan Swift (1667–1745)