projective plane can be defined by using real numbers for the coordinates of their points (Cartesian coordinates for the Euclidean plane and homogeneous coordinates for the projective plane), analogous abstract systems of points and lines can be defined by using other number systems as coordinates. The Sylvester–Gallai theorem does not hold for geometries defined in this way over finite fields: for some finite geometries defined in this way, such as the Fano plane, the set of all points in the geometry has no ordinary lines. The Sylvester–Gallai theorem also does not directly apply to geometries in which points have coordinates that are pairs of complex numbers or quaternions, but these geometries have more complicated analogues of the theorem. For instance, in the complex projective plane there exists a configuration of nine points, Hesse's configuration (the inflection points of a cubic curve), in which every line is non-ordinary, violating the Sylvester–Gallai theorem. Such a configuration is known as a Sylvester–Gallai configuration, and it cannot be realized by points and lines of the Euclidean plane. Another way of stating the Sylvester–Gallai theorem is that whenever the points of a Sylvester–Gallai configuration are embedded into a Euclidean space, preserving colinearities, the points must all lie on a single line, and the example of the Hesse configuration shows that this is false for the complex projective plane. However, proved a complex-number analogue of the Sylvester–Gallai theorem: whenever the points of a Sylvester–Gallai configuration are embedded into a complex projective space, the points must all lie in a two-dimensional subspace. Equivalently, a set of points in three-dimensional complex space whose affine hull is the whole space must have an ordinary line, and in fact must have a linear number of ordinary lines. Similarly, showed that whenever a Sylvester–Gallai configuration is embedded into a space defined over the quaternions, its points must lie in a three-dimensional subspace. Matroids Every set of points in the Euclidean plane, and the lines connecting them, may be abstracted as the elements and flats of a rank-3 oriented matroid. The points and lines of geometries defined using other number systems than the real numbers also form matroids, but not necessarily oriented matroids. In this context, the result of lower-bounding the number of ordinary lines can be generalized to oriented matroids: every rank-3 oriented matroid with elements has at least two-point lines, or equivalently every rank-3 matroid with fewer two-point lines must