point (but not the other points on them) produces the Möbius–Kantor configuration. Partial geometries Given an integer , a tactical configuration satisfying: For every anti-flag there are flags such that and , is called a partial geometry. If there are points on a line and lines through a point, the notation for a partial geometry is . If these partial geometries are generalized quadrangles. If these are called Steiner systems. Generalized polygons For , a generalized -gon is a partial linear space whose incidence graph has the property: The girth of (length of the shortest cycle) is twice the diameter of (the largest distance between two vertices, in this case). A generalized 2-gon is an incidence structure, which is not a partial linear space, consisting of at least two points and two lines with every point being incident with every line. The incidence graph of a generalized 2-gon is a complete bipartite graph. A generalized -gon contains no ordinary -gon for and for every pair of objects (two points, two lines or a point and a line) there is an ordinary -gon that contains them both. Generalized 3-gons are projective planes. Generalized 4-gons are called generalized quadrangles. By the Feit-Higman theorem the only finite generalized -gons with at least three points per line and three lines per point have = 2, 3, 4, 6 or 8. Near polygons For a non-negative integer a near -gon is an incidence structure such that: The maximum distance (as measured in the collinearity graph) between two points is , and For every point and line there is a unique point on that is closest to . A near 0-gon is a point, while a near 2-gon is a line. The collinearity graph of a near 2-gon is a complete graph. A near 4-gon is a generalized quadrangle (possibly degenerate). Every finite generalized polygon except the projective planes is a near polygon. Any connected bipartite graph is a near polygon and any near polygon with precisely two points per line is a connected bipartite graph. Also, all dual polar spaces are near polygons. Many near polygons are related to finite simple groups like the Mathieu groups and the Janko group J2. Moreover, the generalized 2d-gons, which are related to Groups of Lie type, are special cases of near 2d-gons. Möbius planes An abstract Möbius plane (or inversive plane) is an incidence structure where, to avoid