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 **R**3 — 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

### Other articles related to "surface, surfaces":

**Surface**

... The Roman

**surface**or Steiner

**surface**(so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually ... gives parametric equations for the Roman

**surface**as follows x = r2 cos θ cos φ sin φ y = r2 sin θ cos φ sin φ z = r2 cos θ sin θ cos2 φ ... triple point, and each of the xy-, yz-, and xz-planes are tangential to the

**surface**there ...

**Surface**s in Geometry

... Polyhedra, such as the boundary of a cube, are among the first

**surfaces**encountered in geometry ... It is also possible to define smooth

**surfaces**, in which each point has a neighborhood diffeomorphic to some open set in E² ... This elaboration allows calculus to be applied to

**surfaces**to prove many results ...

**Surface**

... The field's

**surface**, originally composed of AstroTurf, contained many gaps and uneven patches ... Baseball players also complained about the

**surface**... It was much harder than other AstroTurf

**surfaces**, and the shock of running on it often caused back pain ...

... In a hard drive, the heads 'fly' above the disk

**surface**with clearance of as little as 3 nanometres ... by the design of an air-bearing etched onto the disk-facing

**surface**of the slider ... role of the air bearing is to maintain the flying height constant as the head moves over the

**surface**of the disk ...

... a set of reference points on the Earth's

**surface**against which position measurements are made and (often) an associated model of the shape of the Earth (reference ellipsoid) to define a geodetic coordinate system ... Horizontal datums are used for describing a point on the Earth's

**surface**, in latitude and longitude or another coordinate system ... In engineering and drafting, a datum is a reference point,

**surface**, or axis on an object against which measurements are made ...

### Famous quotes containing the word surface:

“We tend to be so bombarded with information, and we move so quickly, that there’s a tendency to treat everything on the *surface* level and process things quickly. This is antithetical to the kind of openness and perception you have to have to be receptive to poetry. ... poetry seems to exist in a parallel universe outside daily life in America.”

—Rita Dove (b. 1952)

“Here Men from The Planet Earth

First Set Foot upon The Moon

July, 1969 AD

We Came in Peace for All Mankind”

—Plaque left behind on the moon’s *surface* by the crew of Apollo 11.

“It was a pretty game, played on the smooth *surface* of the pond, a man against a loon.”

—Henry David Thoreau (1817–1862)