The Fano plane is the configuration consisting of the two-dimensional finite projective plane the Galois field of order 2 
. It is not realizable over the real or rational numbers
(Gropp 1997). The incidence structure of the Fano is
plane illustrated above.
It is a block design with 
, 
,
, 
, and 
, the Steiner triple
system 
,
and the unique 
configuration. The Levi
graph of the Fano plane is the Heawood graph.
The connectivity of the Fano plane corresponds to the order-2 two-dimensional Apollonian network.
The Fano plane also solves the Transylvania lottery, which picks three numbers from the integers 1-14. Using
two Fano planes we can guarantee matching two by playing just 14 times as follows.
Label the graph vertices of one Fano plane by the
integers 1-7, the other plane by the integers
8-14. The 14 tickets to play are the 14 lines of the two planes. Then if 
is the winning ticket, at least two of 
are either in the interval [1, 7] or [8, 14]. These two
numbers are on exactly one line of the corresponding
plane, so one of our tickets matches them.
The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24 (Lehmer and Lehmer 1974, Guy 1975, Shanks 1985). The group of automorphisms (incidence-preserving bijections) of the Fano plane is the simple group of group order 168 (Klein 1870).
