Orientability

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

Related Topics:
Surface - Manifold - Orientation (mathematics) - Orientation

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In geometry and topology, a surface in mathbb{R}^3 is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed. Otherwise the surface is said to be orientable.

Related Topics:
Geometry - Topology

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For an abstract surface (i.e., a two-dimensional manifold),

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it is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous (intuitively, locally constant) manner. This turns out be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius band. Thus, for surfaces, the Möbius band may be considered the source of all non-orientability.

Related Topics:
Homeomorphic - Möbius band

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A surface that is embedded in mathbb{R}^3 will be orientable in the letter "R" sense if and only if it is orientable as an abstract surface.

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