In mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy.
In what follows, all surfaces are considered to be second-countable two dimensional manifolds.
- Spheres with n handles attached (called n-tori). These are orientable surfaces with Euler characteristic 2-2n, also called surfaces of genus n.
- Projective planes with n handles attached. These are non-orientable surfaces with Euler characteristic 1-2n.
- Klein bottles with n handles attached. These are non-orientable surfaces with Euler characteristic -2n.
Compact surfaces with boundary are just these with one or more removed disks. A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4.
To make some models, attach the sides of these (and remove the corners to puncture):
* * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B B
sphere real projective plane Klein bottle torus (punctured: Möbius band) (sphere with handle)
This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a manifold.
- Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing