vector space over a skewfield as above. These planes are called Desarguesian planes, named after Girard Desargues. The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called non-Desarguesian planes, and the Moulton plane given above is an example of one. The PG(2, K) notation is reserved for the Desarguesian planes. When K is a field, a very common case, they are also known as field planes and if the field is a finite field they can be called Galois planes. Subplanes A subplane of a projective plane is a subset of the points of the plane which themselves form a projective plane with the same incidence relations. proves the following theorem. Let Π be a finite projective plane of order N with a proper subplane Π0 of order M. Then either N = M2 or N ≥ M2 + M. When N is a square, subplanes of order are called Baer subplanes. Every point of the plane lies on a line of a Baer subplane and every line of the plane contains a point of the Baer subplane. In the finite Desarguesian planes PG(2, pn), the subplanes have orders which are the orders of the subfields of the finite field GF(pn), that is, pi where i is a divisor of n. In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of order M in a plane of order N with M2 + M = N is an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order. Fano subplanes A Fano subplane is a subplane isomorphic to PG(2, 2), the unique projective plane of order 2. If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called the diagonal points of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle). In finite desarguesian planes, PG(2, q), Fano subplanes