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 ${\displaystyle \mathbb {R} ^{3}}$ passing through the origin.

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.