{| class=wikitable align=right | valign=top width=120|120pxthe fundamental polygon of the projective plane. | valign=top width=120|120pxthe mÃ¶bius strip with a single edge, can be closed into a projective plane by gluing opposite open edges together. | valign=top width=120|120pxin comparison the klein bottle is a mÃ¶bius strip closed into a cylinder. |} in mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, 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 r3 passing through the origin. the plane is also often described topologically, in terms of a construction based on the mÃ¶bius strip: if one could glue the (single) edge of the mÃ¶bius strip to itself in the correct direction, one would obtain the projective plane. (this cannot be done in three-dimensional space.) equivalently, gluing a disk along the boundary of the mÃ¶bius strip gives the projective plane. topologically, it has euler characteristic 1, hence a demigenus (non-orientable genus, euler genus) of 1. since the mÃ¶bius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square (that is, [0,1] &times; [0,1] ) with its sides identified by the following equivalence relations: :(0, y) ~ (1, 1 &minus; y) &nbsp; for 0&nbsp;&le;&nbsp;y&nbsp;&le;&nbsp;1 and :(x, 0) ~ (1 &minus; x, 1) &nbsp; for 0&nbsp;&le;&nbsp;x&nbsp;&le;&nbsp;1, as in the leftmost diagram on the right. == examples == projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the euclidean plane or 3-space in many different ways.ap&eacute;ry, f.; models of the real projective plane, vieweg (1987) some of the more important examples are described below. the projective plane cannot be embedded (that is without intersection) in three-dimensional euclidean space. the proof that the projective plane does not embed in three-dimensional euclidean space goes like this: assuming that it does embed, it would bound a compact region in three-dimensional euclidean space by the generalized jordan curve theorem. the outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be the projective plane, which is not orientable. this is a 
Geometric_topology Surfaces