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.

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.