Fano plane

These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements.

The standard notation for this plane, as a member of a family of projective spaces, is PG(2, 2).

Using the standard construction of projective spaces via homogeneous coordinates, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111.

The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits.

As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident.

F8 is a degree-three field extension of F2, so the points of the Fano plane may be identified with F8 ∖ {0}.

You will obtain a complete graph on seven vertices with seven colored triangles (projective lines).

[5] Dualities can be viewed in the context of the Heawood graph as color reversing automorphisms.

An example of a polarity is given by reflection through a vertical line that bisects the Heawood graph representation given on the right.

This is also an immediate consequence of the symmetry between points and lines in the definition of the incidence relation in terms of homogeneous coordinates, as detailed in an earlier section.

colors can be calculated by plugging the numbers of cycle structures into the Pólya enumeration theorem.

A famous result, due to Andrew M. Gleason states that if every complete quadrangle in a finite projective plane extends to a Fano subplane (that is, has collinear diagonal points) then the plane is Desarguesian.

To compound the confusion, Fano's axiom states that the diagonal points of a complete quadrangle are never collinear, a condition that holds in the Euclidean and real projective planes.

[9] The Fano plane contains the following numbers of configurations of points and lines of different types.

[13] The Fano plane is a small symmetric block design, specifically a 2-(7, 3, 1)-design.

[14] With the lines labeled ℓ0, ..., ℓ6 the incidence matrix (table) is given by: The Fano plane, as a block design, is a Steiner triple system.

The Fano plane can be extended in a third dimension to form a three-dimensional projective space, denoted by PG(3, 2).

It has 15 points, 35 lines, and 15 planes and is the smallest three-dimensional projective space.