December 29, 2016

... is shorthand for the projective plane of order ... . The first figure presents ... ), the best-known 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 ... . Color-coded 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. Multi-point lines can be seen by selecting indices ... and ... . When a failure is reported, exploration reveals cases with multiple (or no) intersections.