Projective Planes of Low Order

Fano

center

{0,0}

{0.3,0}

{0,0.3}

{-0.2,0.3}

PP(n)

is shorthand for the projective plane of order

. The first figure presents

PP(2

), 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

n+1=3

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

n=2

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

n=1,2,3

. 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

PP(n)

, (if it exists) is a pair of sets of

's and

s such that any two

's determine exactly one

, while

n+1

'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

n+1

points lie on each line, and

n+1

lines pass through each point.

There must be

+n+1

points (and lines) in

PP(n)

. This Demonstration uses a simple algorithm that only creates

PP(n)

for prime

. It is too slow for

n>10

Color-coded regular graphs are created and shown; each colored line is a polygon of

n+1

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

n>3

. 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

n=6

n=10

, by the Bruck–Ryser–Chowla theorem and by exhaustive computation, respectively. The status for

n=12

has not been established. Another theorem states that

PP(n)

exists if

is a prime power. Published results are used to show

PP(4)

PP(8)

, and

PP(9)

, 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.

Details

The thumbnail is

PP(2)

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

PP(2)

, with each "line" expanded into a triangle.

PP(4)

(the first prime-power case) is shown similarly. The last two snapshots show an invalid attempt to create

PP(6)

, with lines sharing two points.

The

PP(4)

PP(8)

, and

PP(9)

data are adapted from Projective Planes of Small Order.

Projective planes are restricted cases of block designs

BD[v,k,λ]

, with

vertices, partitioned into blocks (sets of

vertices) in such a way that any two vertices are in exactly

blocks.

PP(n)

(if it exists) is

BD[v=

+n+1,k=n+1,λ=1]

. A configuration

points each on exactly

lines; points and lines can be at infinity; only

PP(0)

PP(1)

, and

PP(2)

allow a configuration.

BD[7,3,1]

, configuration

, and the Fano plane all define

PP(2)

External Links

Fano Plane (Wolfram MathWorld)

Projective Plane (Wolfram MathWorld)

Block Design (Wolfram MathWorld)

Configuration (Wolfram MathWorld)

Permanent Citation

Roger Beresford

"Projective Planes of Low Order"
http://demonstrations.wolfram.com/ProjectivePlanesOfLowOrder/
Wolfram Demonstrations Project
Published: January 29, 2010