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# Projective Planes of Low Order

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.

### DETAILS

The thumbnail is as the Fano plane with seven "lines", and shows features that are not typical of other projective planes. The first snapshot is a more representative version of , with each "line" expanded into a triangle. (the first prime-power case) is shown similarly. The last two snapshots show an invalid attempt to create , with lines sharing two points.
The , , and data are adapted from Projective Planes of Small Order.
Projective planes are restricted cases of block designs , with vertices, partitioned into blocks (sets of vertices) in such a way that any two vertices are in exactly blocks. (if it exists) is . A configuration is points each on exactly lines; points and lines can be at infinity; only , , and allow a configuration. , configuration , and the Fano plane all define .

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