exist if and only if q is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders). An open question, apparently due to Hanna Neumann though not published by her, is: Does every non-desarguesian plane contain a Fano subplane? A theorem concerning Fano subplanes due to is: If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order). Affine planes Projectivization of the Euclidean plane produced the real projective plane. The inverse operation—starting with a projective plane, remove one line and all the points incident with that line—produces an affine plane. Definition More formally an affine plane consists of a set of lines and a set of points, and a relation between points and lines called incidence, having the following properties: Given any two distinct points, there is exactly one line incident with both of them. Given any line l and any point P not incident with l, there is exactly one line incident with P that does not meet l. There are four points such that no line is incident with more than two of them. The second condition means that there are parallel lines and is known as Playfair's axiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines." The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2, q) are denoted by AG(2, q). There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N, but the converse is not true. The affine planes formed by the removal of different