Orientability

:This article discusses orientability and orientation on surfaces and manifolds. For orientation of vector spaces see orientation (mathematics). For alternate uses, see orientation.

Examples in low dimensions

Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces are Möbius strip, real projective plane, Klein bottle. These surfaces as visualized in 3-dimensions all have just one-side. 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. (Caveat: the real projective plane and Klein bottle can't be embedded in mathbb R^3, only immersed with nice intersections.)

Related Topics:
Sphere - Plane - Torus - Möbius strip - Real projective plane - Klein bottle

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In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as mathbb{R}^3 above) is orientable. For example, a torus embedded in K^2 imes 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.

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