in the Euclidean plane using only points and straight line segments (i.e., it is not realizable). This is a consequence of the Sylvester–Gallai theorem, according to which every realizable incidence geometry must include an ordinary line, a line containing only two points. The Fano plane has no such line (that is, it is a Sylvester–Gallai configuration), so it is not realizable. A complete quadrangle consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the Fano axiom, often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear. Affine planes An affine plane is a linear space satisfying: For any point and line not incident with it (an anti-flag) there is exactly one line incident with (that is, ), that does not meet (known as Playfair's axiom), and satisfying the non-degeneracy condition: There exists a triangle, i.e. three non-collinear points. The lines and in the statement of Playfair's axiom are said to be parallel. Every affine plane can be uniquely extended to a projective plane. The order of a finite affine plane is , the number of points on a line. An affine plane of order is an configuration. Hesse configuration The affine plane of order three is a configuration. When embedded in some ambient space it is called the Hesse configuration. It is not realizable in the Euclidean plane but is realizable in the complex projective plane as the nine inflection points of an elliptic curve with the 12 lines incident with triples of these. The 12 lines can be partitioned into four classes of three lines apiece where, in each class the lines are mutually disjoint. These classes are called parallel classes of lines. Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any single line and all the points on that line produces this affine plane of order three (it is also unique). Removing one point and the four lines that pass through that