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).
." Comment. Math. Helv. 35, 85-92, 1961.Pinkall,
U.