Editing Incidence geometry (section)

====Hesse configuration====
{{main|Hesse configuration}}

The affine plane of order three is a {{math|(9<sub>4</sub>, 12<sub>3</sub>)}} 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 point]]s 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 {{math|(13<sub>4</sub>)}} 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 point (but not the other points on them) produces the {{math|(8<sub>3</sub>)}} [[Mobius-Kantor configuration|Möbius–Kantor configuration]].