Projective Planes of Low Order
Projective Planes of Low Order
PP(n)
n
PP(2
n+1=3
n
a
b
n=2
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.
n
n=1,2,3
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.
a
b
a
b
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.
n
PP(n)
p
q'
p
q
n+1
p
q
p
q
p
q
n+1
n+1
There must be +n+1 points (and lines) in . This Demonstration uses a simple algorithm that only creates for prime . It is too slow for .
2
n
PP(n)
PP(n)
n
n>10
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.
n+1
n>3
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.
n
n=6
n=10
n=12
PP(n)
n
PP(4)
PP(8)
PP(9)
a
b