# Projective Planes of Low Order

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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.

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Contributed by: Roger Beresford (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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 .

## Permanent Citation