Real Projective Plane

The real projective plane is the closed topological manifold, denoted , that is obtained by projecting the points of a plane from a fixed point (not on the plane), with the addition of the line at infinity. It can be described by connecting the sides of a square in the orientations illustrated above (Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324).

There is then a one-to-one correspondence between points in and lines through not parallel to . Lines through that are parallel to have a one-to-one correspondence with points on the line at infinity. Since each line through intersects the sphere centered at and tangent to in two antipodal points, can be described as a quotient space of by identifying any two such points. The real projective plane is a nonorientable surface. The equator of (which, in the quotient space, is itself a projective line) corresponds to the line at infinity.

The complete graph on 6 vertices can be drawn in the projective plane without any lines crossing, as illustrated above. Here, the projective plane is shown as a dashed circle, where lines continue on the opposite side of the circle. The dual of on the projective plane is the Petersen graph.

The Boy surface, cross-cap, and Roman surface are all homeomorphic to the real projective plane and, because is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986).

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