is shorthand for the projective plane of order The first figure presents the bestknown finite projective plane the Fano plane with 7 points on 7 lines The central triangle often drawn as a circle is the seventh line Each point lies on lines and each line also passes through 3 points; every pair of points defines a single line and every pair of lines defines a single point This presentation is shown when Fano is selected It does not generalize to higher orders because it is a configuration where points can be at the end or middle of a line The controls center and do not apply in this case There is no difference between the two representations for or Fano except a rearrangement of the lines Selecting an integer value of gives an abstract projective plane in which concepts such as between middle and end are undefined Look at Change the center to reveal hidden lines The controls and let you see individual lines and check that pairs share just one point restore and to 0 afterwards Then read the following definition The projective plane of order if it exists is a pair of sets of s and s such that any two s determine exactly one while s relate to each ; duality means that these statements are still true after exchanging and The s and ’s are often called points and lines; the relationships are then that points lie on each line and lines pass through each point There must be points and lines in This Demonstration uses a simple algorithm that only creates for prime It is too slow for Colorcoded regular graphs are created and shown; each colored line is a polygon of points and includes one point of the same color A more accurate representation would use a complete graph for each line with relationships shown as edges between every point in the line but this would be illegible for The central point has no special significance; all points are equal Not all values of give rise to finite projective planes; it is not always possible to restrict pairs of points to single lines Projective planes have been proven not to exist for or by the Bruck–Ryser–Chowla theorem and by exhaustive computation respectively The status for has not been established Another theorem states that exists if is a prime power Published results are used to show and for which my algorithm fails A test checks whether any pairs of points lie on more than one line reporting the first failure Multipoint lines can be seen by selecting indices and When a failure is reported exploration reveals cases with multiple or no intersections