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.