Editing Orientability (section)

==Orientable surfaces==
[[File:Surface orientation.gif|thumb|300px|right|In this animation, a simple analogy is made using a gear that rotates according to the right-hand rule on a surface's normal vector. The orientation of the curves given by the boundaries is given by the direction in which the dots move as they are pushed by the moving gear. On a non-orientable surface, such as the Möbius strip, the boundary would have to move in both directions at once, which is not possible.]]

A surface ''S'' in the [[Euclidean space]] '''R'''<sup>3</sup> is '''''orientable''''' if a [[Chirality (mathematics)|chiral]] two-dimensional figure (for example, [[File:Small pie.svg|20px]]) cannot be moved around the surface and back to where it started so that it looks like its own mirror image ([[File:pie 2.svg|20px]]). Otherwise the surface is '''''non-orientable'''''.  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.  That is to say that a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to a loop going around the opposite way. 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.

For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an '''''orientation''''', and the surface is called '''''oriented'''''.  For surfaces embedded in Euclidean space, an orientation is specified by the choice of a continuously varying [[surface normal]] '''n''' at every point.  If such a normal exists at all, then there are always two ways to select it: '''n''' or &minus;'''n'''.  More generally, an orientable surface admits exactly two orientations, and the distinction between an orient''ed'' surface and an orient''able'' 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 two possible orientations.

===Examples===
Most surfaces encountered in the physical world are orientable.  [[Sphere]]s, [[plane (mathematics)|planes]], and [[torus|tori]] are orientable, for example.  But [[Möbius strip]]s, [[real projective plane]]s, and [[Klein bottle]]s are non-orientable. They, as visualized in 3-dimensions, all have just one side.  The real projective plane and Klein bottle cannot be embedded in '''R'''<sup>3</sup>, only [[immersion (mathematics)|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 '''R'''<sup>3</sup>  above) is orientable. For example, a torus embedded in 
:<math>K^2 \times S^1</math>

can be one-sided, and a Klein bottle in the same space can be two-sided; here <math>K^2</math> refers to the Klein bottle.

===Orientation by triangulation===
Any surface has a [[triangulation (topology)|triangulation]]: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. Each triangle is oriented by choosing a direction around the perimeter of the triangle, associating a direction to each edge of the triangle.  If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface.  Such a choice is only possible if the surface is orientable, and in this case there are exactly two different orientations.

If the figure [[File:Small pie.svg|20px]] can be consistently positioned at all points of the surface without turning into its mirror image, then this will induce an orientation in the above sense on each of the triangles of the triangulation by selecting the direction of each of the triangles based on the order red-green-blue of colors of any of the figures in the interior of the triangle.

This approach generalizes to any ''n''-manifold having a triangulation.  However,  some 4-manifolds do not have a triangulation, and in general for ''n'' > 4  some ''n''-manifolds have triangulations that are inequivalent.

===Orientability and homology===

If ''H''<sub>1</sub>(''S'') denotes the first [[Homology (mathematics)|homology]] group of a closed surface ''S'', then ''S'' is orientable if and only if ''H''<sub>1</sub>(''S'') has a trivial [[torsion subgroup]]. More precisely, if ''S'' is orientable then ''H''<sub>1</sub>(''S'') is a [[free abelian group]], and if not then ''H''<sub>1</sub>(''S'') = ''F'' + '''Z'''/2'''Z''' where ''F'' is free abelian, and the '''Z'''/2'''Z''' factor is generated by the middle curve in a [[Möbius band]] embedded in ''S''.