# Real projective plane

## Compact non-orientable two-dimensional manifold / From Wikipedia, the free encyclopedia

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In mathematics, the **real projective plane** is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in $\mathbb {R} ^{3}$ passing through the origin.

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The fundamental polygon of the projective plane. |
The Möbius strip with a single edge, can be closed into a projective plane by gluing opposite open edges together. |
In comparison, the Klein bottle is a Möbius strip closed into a cylinder. |

The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.

Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together with a half-twist, the *real* projective plane can thus be represented as a unit square (that is, [0, 1] × [0,1]) with its sides identified by the following equivalence relations:

- (0,
*y*) ~ (1, 1 −*y*) for 0 ≤*y*≤ 1

and

- (
*x*, 0) ~ (1 −*x*, 1) for 0 ≤*x*≤ 1,

as in the leftmost diagram shown here.