A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself. Equivalently, the real projective plane can be represented by a closed disk in which each pair of antipodal points on the boundary is identified.
Topologically, the real projective plane is a nonorientable surface with Euler characteristic 1, in contrast to the sphere, which is an orientable surface with Euler characteristic 2. For graph embeddings, drawing on the sphere is equivalent to drawing in the plane, since a point of the sphere not on the graph can be removed and the remainder flattened into the plane. The real projective plane provides a genuinely different surface for graph embeddings.
Every planar graph also embeds on the real projective plane, since a planar embedding
can be placed inside a disk on that surface.
The converse is false: the complete graph 
and the complete
bipartite graph 
are projective planar graphs but are not
planar graphs. Placing a planar
embedding in such a disk gives a valid graph
embedding, but it need not give a cellular
embedding, since the complementary region containing the cross-cap
is not itself a disk. These graph
embeddings on the projective plane refer
to this topological real projective plane,
not to the finite incidence structures defined
below.
A finite projective plane of order 
is formally defined as a set of 
points with the properties
that:
1. Any two points determine a line,
2. Any two lines determine a point,
3. Every point has 
lines on it, and
4. Every line contains 
points.
(Note that some of these properties are redundant.) A projective plane is therefore a symmetric (
, 
, 1) block design. An affine plane of order 
exists iff a projective plane of order
exists.
A finite projective plane exists when the order 
is a power of a prime,
i.e., 
for 
.
It is conjectured that these are the only possible projective planes, but
proving this remains one of the most important unsolved problems in combinatorics.
The first few orders that are powers of primes are 2, 3, 4, 5, 7, 8, 9, 11,
13, 16, ... (OEIS A000961). The first few orders
that are not of this form are 6, 10, 12, 14, 15, ... (OEIS A024619).
The smallest finite projective plane is of order 
, and consists of the 
configuration known as
the Fano plane, illustrated above.
The remarkable Bruck-Ryser-Chowla theorem says that if a projective plane of order 
exists, and 
or 2 (mod 4), then 
is the sum of two squares.
This rules out 
.
By answering Lam's problem in the negative using
massive computer calculations on top of some mathematics, it has been proved that
there are no finite projective planes of order 10 (Lam 1991). The status of the order
12 projective plane remains open.
The projective plane of order 2, also known as the Fano plane, is denoted PG(2, 2). It has incidence matrix
![[1 1 1 0 0 0 0; 1 0 0 1 1 0 0; 1 0 0 0 0 1 1; 0 1 0 1 0 1 0; 0 1 0 0 1 0 1; 0 0 1 1 0 0 1; 0 0 1 0 1 1 0].](/images/equations/ProjectivePlane/NumberedEquation1.svg) |
Every row and column contains 3 1s, and any pair of rows/columns has a single 1 in common.
The projective plane has Euler characteristic 1, and the Heawood conjecture therefore shows that any set of regions on it can be colored using six colors only (Saaty 1986). The Petersen graph provides a 6-color coloring of the projective plane.
