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,
, 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.
let you see individual lines and check that pairs share just one point (restore
to 0 afterwards). Then read the following definition.
The projective plane of order
, (if it exists) is a pair of sets of
s such that any two '
s determine exactly one
's "relate" to each
; duality means that these statements are still true after exchanging
’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
. 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
, by the Bruck–Ryser–Chowla theorem and by exhaustive computation, respectively. The status for
has not been established. Another theorem states that
is a prime power. Published results are used to show
, 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
. When a failure is reported, exploration reveals cases with multiple (or no) intersections.