For: orientation (mathematics)

More precisely, and applicable to non-embedded surfaces, a surface is orientable if there is no continuous map f from the product of a 2-dimensional ball B and the unit interval 1 to the surface, $f:\; B\backslash times2\backslash to\; S$ such that f(b,t)=f(c,t) only if b=c for every t in 3, and there exists a reflection map r such that f(b,0) = f(r(b),1) for every b in B.

An abstract surface (i.e., a two-dimensional manifold) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. This turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability.

A surface that is embedded in R³ will be orientable in the it is orientable as an abstract surface.

Oriented vs. orientable

For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an orientation, and the surface is called oriented. An orientable surface admits exactly 2 orientations, and the distinction between an oriented surface and an orientable surface is subtle and frequently blurred. An orientable surface is an abstract surface that admits an orientation, while an oriented surface is a surface that is abstractly orientable, and has the additional datum of a choice of one of the 2 possible orientations.

Examples

Most surfaces we encounter in the physical world are orientable. Spheres, planes, and tori are orientable, for example. But Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side. (Caveat: the real projective plane and Klein bottle can't be embedded in R³, only immersed with nice intersections.)

Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough.

In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as R³ above) is orientable. For example, a torus embedded in $K^2\; \backslash times\; S^1$ can be one-sided, and a Klein bottle in the same space can be two-sided; here $K^2$ refers to the Klein bottle.