finite projective plane

finite projective plane

Finite projective planes from axioms

Let us start from the beginning and assume we are given a set of

n

things called

point

s and a set of

ν

things called

line

s, with an

incidence relation

between them (a

point

is

incident

with a

line

iff the

line

is incident with the

point

) that

satisfies

only two

axioms

, the two unique incidence properties:

•

For any two distinct

point

s,
there is exactly one

line

through both of them.

•

For any two distinct

line

s,
there is exactly one

point

on both of them.

Given these

http://planetmath.org/node/6940

projective plane

axioms we can apply the De Bruijn–Erdős

theorem

twice (once with the rôle of

point

s and

line

s swapped) which tells us we must have

n

=

ν

, and one of two possible

configurations

:

∘

One “degenerate”

solution

has one special line

ℓ

0

and

n

-

1

ordinary lines

ℓ

i

, and a special point
P

0

and

n

-

1

ordinary points

P

i

. Each P

i

(

i

≠

0

) is
incident with its own

ℓ

i

and with

ℓ

0

; each

ℓ

i

(

i

≠

0

) with its own P

i

and with P

0

. Now

ℓ

0

is
incident with every point except

ℓ

0

while P

0

is
incident with every line except

ℓ

0

.

∘

The other solution is a projective plane:

∙

the number of

point

s and the number of

line

s are of the form

q

2

+

q

+

1

for some

integer

q

;

∙

each

point

is incident with

q

+

1

line

s and each

line

with

q

+

1

point

s.

One usually avoids the degenerate case by demanding there is a

quadrangle

in the plane: four

point

s no three of which lie on the same

line

. This, or a

similar

non-triviality clause, can then be adopted as a third axiom.
Unfortunately it also has the effect of excluding projective planes of orders

q

=

0

and

1

. The
De Bruijn–Erdös (

http://planetmath.org/DeBruijnErdHosTheorem

)
theorem actually makes an additional

assumption

: that
no two lines are incident with the same set of points (and when applied dually
that no two points are incident with the same set of lines). That is not a
problem for a

line

incident with more than one

point

or vice versa, because
then the axioms forbid it. We just get some more degenerate cases consisting
of

multiple

lines intersecting one point or none, or vice versa; the insistence
on a quadrangle catches these too.

I will not prove the whole theorem here, but under the assumption
that a quadrangle exists it is not hard show the interesting part,
that those numbers

q

+

1

and

q

2

+

q

+

1

drop out by

necessity

.
The same construction can also be used to give

coordinates

to the

point

s and

line

s of the plane (see below).

Projective planes as bipartite graphs

Projective planes are, apart from anything else, also

bipartite graphs

: let the

q

2

+

q

+

1

point

s be represented by “black” nodes (vertices) of the graph,
and let the

q

2

+

q

+

1

line

s

also

be nodes, “white” ones say.
Edges will

represent

incidence between a “black” and a “white” node,
so there are

(

q

2

+

q

+

1

)

⁢

(

q

+

1

)

=

q

3

+

2

⁢

q

2

+

2

⁢

q

+

1

edges in all.

•

In a bipartite graph, nodes of the same color have even

distance

.
This is 0 for any node and itself and

must be 2

for any two
distinct nodes of the same color: for any two

point

s there is a path
via the

line

they share, and for any two

line

s a path via the

point

they share. So distance 4 does not occur.

For nodes of opposite color we can have 1 (if the

point

lies on the

line

) or 3 (otherwise); we can’t have distance 5 which would imply
4 for some nodes. So the graph has

diameter

(maximum distance) 3.

•

In a bipartite graph, all cycles (if any) have even length. We
don’t have 2-cycles (this is not a

multigraph

) but no 4-cycles
either, by the following

argument

: such a cycle P

ℓ

Q

m

would mean two

line

s intersecting in two different

point

s, and
vice versa. So the

girth

(minimum cycle length) is 6
(showing there indeed are 6-cycles for

q

⩾

2

is left as
exercise for the reader).

These two conditions, diameter 3 and girth 6, are not only necessary for
the graph to represent a projective plane, they are also sufficient, and therefore form
an alternative formulation of what it means to be a projective plane.

These graphs could be called the bipartite analogue of

Moore graphs

of
diameter 2 and girth 5; many of the same arguments can be used

[

vG03

]

.

Coördinates for finite projective planes

The details on how to impose coordinates on a projective plane can be found in

[

Hal59

,

dRe96

]

, or in this entry (

http://planetmath.org/IntroducingCoordinatesInAProjectivePlane

). We summarize it for a

finite projective plane

below:

Setup: Use a set

Q

of

q

different symbols for coordinate values, two of which are 0 and 1. Start with four

point

s O, I, X, Y forming a quadrangle. XY is the “

line

at

infinity

”

ℓ

.

Construction: O and I are given coordinates

(

0

,

0

)

and

(

1

,

1

)

, and the other

point

s on OI (except its

intersection

with

ℓ

) arbitrarily

(

c

,

c

)

with unique

c

∈

Q

. For any

point

P not on

ℓ

, let YP
intersect OI at

(

a

,

a

)

, and let XP intersect OI at

(

b

,

b

)

. Then P will be given

(

a

,

b

)

. For the coordinates of

point

s on

ℓ

, intersect the

line

s joining

(

0

,

0

)

with

(

1

,

m

)

, for all

m

∈

Q

, with

ℓ

, calling the intersection

(

m

)

. Finally, Y gets coordinate

(

)

(instead of

(

∞

)

in the literature).

Lines

not through Y are given coordinates

[

m

,

k

]

where

(

m

)

and

(

0

,

k

)

are their intersections with

ℓ

and with OY respectively;

line

s through Y (other than

ℓ

) get coordinates

[

k

]

going by their
intersection

(

k

,

0

)

with OX. The

line

ℓ

gets

[

]

(instead of

[

∞

]

in the literature).

Coordinates from bipartite graphs

The whole business with coordinates becomes a bit more transparent when we try
to actually

construct

the thing as a bipartite graph, starting from one edge,
Y

ℓ

say.

There are

q

more

point

s (other than Y) incident to

ℓ

, let’s call
them Y

m

where

m

ranges over the

q

different values of the set

Q

.
Again, each Y

m

is incident to

q

more

line

s (apart from

ℓ

), let’s
call those

ℓ

m

⁢

b

where

b

is another

subscript

from

Q

.
Likewise there are

q

more

line

s (other than

ℓ

) incident to Y, let’s
call them

ℓ

x

and to each of them are incident

q

more

point

s (apart
from Y) which we can label Y

x

⁢

y

.

This immediately produces the incidence list of the preceding

section

.
Pictorially, for

q

=

2

(with the “to be assessed” portion filled in
as thinner lines):

We already have

ℓ

0

containing all points Y

0

⁢

y

, and in

addition

if we
wanted we could easily arrange for

ℓ

00

to be incident with all the
Y

x

⁢

0

(just renumber the labels

y

in the sets Y

x

⁢

0

for every

x

),
and for

ℓ

10

to be incident with all the Y

c

⁢

c

(just look at which
Y

x

⁢

y

are incident with

ℓ

10

, then exercise the freedom we still have
to relabel all the

x

,

matching

each time the corresponding

y

there). This
reproduces the entire coordinatisation.

Projective planes as permutation systems

Five sides of the

hexagon

are self-evident. On the left we have the
single edge Y

ℓ

, along the top we fan out in two stages to

q

and

q

2

items, along the bottom we also fan out in two stages to

q

and

q

2

items,
and the hard part is the right. We have by now accounted for

2

⁢

q

2

+

2

⁢

q

+

1

edges
so we know there must be exactly

q

3

more edges, between

some

of
those

q

2

point

s and

some

of those

q

2

line

s. But where?

For

q

=

2

it is simple enough to avoid 4-cycle clashes (and the

diagram

shows
what is, up to

isomorphism

, the only solution). In general, it is an

open
problem

to find all possible solutions.

From the projective plane properties it is easy to see that any point Y

x

⁢

y

must, for
every value of

m

, be incident to no more than one of the

ℓ

m

⁢

b

(because
if it was incident to two they would intersect both in our Y

x

⁢

y

and in
their shared Y

m

at the bottom). On the other hand, Y

x

⁢

y

needs

q

more
mates so it needs to choose exactly one

ℓ

m

⁢

b

for every value of

m

.

By exactly the same reasoning, any

ℓ

m

⁢

b

needs to be incident with, for
every value of

x

, at most one and then exactly one Y

x

⁢

y

. In other words,
the projective plane

structure

takes the form of, for each pair of values

x

and

m

,
a

bijection

ψ

x

⁢

m

between the

y

indices and the

b

indices.

This is not yet the only constraint. A closer

analysis

[

vG03

]

shows
that, in order to prevent any two

line

s meeting in more than one

point

or any
two

point

s in more than two

line

s, all compound bijections of the form

ψ

x

⁢

m

⁢

ψ

x

′

⁢

m

-

1

⁢

ψ

x

′

⁢

m

′

⁢

ψ

x

⁢

m

′

-

1

(for

m

≠

m

′

and

x

≠

x

′

) must be

derangements

, i.e.

permutations

that do not leave any index in place. It is not hard to see
this is just a device to prevent 4-cycles. And careful scrutiny of the graph
shows that, after arranging 5 of 6 edges along trees fanning out as above, and
after recognising there are those bijections along the sixth edge, these
derangement shenanigans stop the only remaining gap for 4-cycles to lurk.

This could be called the

permutation group

interpretation

of projective plane structure.
The

algebraic

version appears in the next section.

Planar ternary rings

We’ve been giving our

point

s and

line

s cooordinates from a set

Q

of

q

different labels. We can give

Q

the structure of a

ternary ring

, that is, a set with a ternary operation

p

∙

q

∘

r

, by defining

y

=

x

∙

m

∘

b

⟺

(

x

,

y

)

⁢

is on

⁢

[

m

,

b

]

(1)

Thus, geometric properties of a projective plane may not be analyzed algebraically. See this entry (

http://planetmath.org/TernaryRingOfAProjectivePlane

) for more detail.

Conversely

, every ternary ring determines a projective plane with coordinates as above, so that algebraic properties of the ring can be interpreted geometrically. For more detail on this, see this entry (

http://planetmath.org/ProjectivePlaneOfATernaryRing

).

Fields and division rings are examples of ternary rings. But there are other classes of

algebraic structures

intermediate between fields and ternary rings. See

[

dRe96

]

for a survey. There are four different projective planes known of order 9, many more of order 16, and so on.

Finding all PTRs in general remains of course just as hard as finding projective planes!

Finite projective planes as incidence structures

Finite projective planes can be defined as
(

simple square

) 2-

(

q

2

+

q

+

1

,

q

+

1

,

1

)

designs

,
or what’s the same thing,

Steiner systems

S

⁢

(

2

,

q

+

1

,

q

2

+

q

+

1

)

.
This way, we already

assume

in the definition that each “block”
(

line

) contains exactly

q

+

1

point

s, from which it follows by calculation that
the number of

line

s that contain a given

point

is also

q

+

1

, and that the
total number of

line

s equals the total number of

point

s.

Also, the unique incidence property that every two

point

s lie together on
exactly one

line

is given, and the dual one that every two

line

s meet in
exactly one

point

then follows from the numbers and the properties that define
a design or Steiner system.

We took a different route to projective planes above: we saw we only need to start with
these two unique incidence properties, and can

prove

from them that
the number of

point

s on a

line

and the number of

line

s through a

point

are both
fixed (i.e. that the projective plane and its dual are both designs).

Finite projective planes as incidence matrices

Consider an arbitrary projective plane of order

q

.
We know there are

n

=

q

2

+

q

+

1

points and the same number of lines. Let

A

be an

n

×

n

matrix of which the rows and columns represent lines and
points of the plane, and give the value 1 to entries at intersections where
the point (represented by the column) lies on the line (represented by the
row), and leave the other entries zeros. Such an

A

is known as the

incidence matrix

of the plane.

A

⁢

A

⊤

has rows and columns both representing lines, and the entries
here are the

dot products

of any two rows of

A

, i.e. the number of points
any two lines

l

and

m

have in common, which is

q

+

1

when

l

=

m

and 1
otherwise, by the projective plane properties.

A

⁢

A

⊤

=

(

q

+

1

1

1

⋯

1

q

+

1

1

⋯

1

1

q

+

1

⋯

⋮

⋮

⋮

⋱

)

=

J

+

q

⁢

I

where

I

is the

identity matrix

(ones along the

diagonal

, zeros everywhere
else) and

J

is the matrix filled with ones in every entry. Note

A

⁢

A

⊤

has this same value for

every

projective plane with

n

points and

n

lines,
while

A

on its own of course varies if there is more than one plane of the
same order, as the plane is fully determined by its incidence matrix. Also,

A

⁢

A

⊤

having this value is the

only

thing needed to make the plane
specified by

A

satisfy the projective plane properties; it is a fully

equivalent

formulation of those properties.

The

determinant

of

A

is best evaluated via that of

A

⁢

A

⊤

,
because

(

det

⁡

A

)

2

=

(

det

⁡

A

)

⁢

(

det

⁡

A

⊤

)

=

det

⁡

A

⁢

A

⊤

, and we know
every entry of

A

⁢

A

⊤

. The

eigenvalues

of any

n

×

n

matrix of
the form

J

+

q

⁢

I

are easily seen to be

•

q

+

n

(once) with

eigenvector

𝐱

for which

x

0

=

x

1

=

x

2

⁢

…

•

q

(

multiplicity

n

-

1

) for the

complementary

eigenspace

x

0

+

x

1

+

x

2

⁢

⋯

=

0

by letting the matrix act on these eigenvectors. This gives

det

⁡

A

⁢

A

⊤

=

q

n

-

1

⁢

(

q

+

n

)

which in our case (

n

=

q

2

+

q

+

1

) amounts to

det

⁡

A

⁢

A

⊤

=

q

q

2

+

q

⁢

(

q

2

+

2

⁢

q

+

1

)

As

q

2

+

q

is even and the other factor a square, we get an integer

det

⁡

A

=

±

q

q

2

+

q

2

⁢

(

q

+

1

)

=

±

q

(

q

+

1

2

)

⁢

(

q

+

1

)

where the sign is moot, as

A

has no correspondence between its columns and
rows.

Existence

The

Bruck–Ryser theorem

[

BR49

]

states that

If

q

is the order of a projective plane, and

q

≡

1

or

2

⁢

(

mod

⁢

4

)

, then

q

is
the sum of two integer squares (one of which can be

0

2

).

It can be proven

[

BR49

,

Cam94

]

using the incidence matrix

A

, by a
clever rearrangement of

𝐱

⁢

A

⁢

A

⊤

⁢

𝐱

(plus one extra term) for an
arbitrary vector

𝐱

, on which gradually constraints are imposed four

components

at a time using the fact that each integer is a sum of four
squares. The argument only works when

q

2

+

q

+

1

=

n

≡

-

1

⁢

(

mod

⁢

4

)

, hence the
constraint on

q

(mod 4).

Interestingly, there is a partial

converse

[

HR54

]

: whenever

q

is
allowed by the theorem (either

congruent

0 or 3 (mod 4), or congruent 1
or 2 (mod 4) and a

sum of two squares

), there always is a

rational

n

×

n

matrix

A

(where

n

=

q

2

+

q

+

1

) such that

A

⁢

A

⊤

=

M

. Of course,
the matrix is only a proper incidence matrix if that rational matrix only has
entries 0 and 1.

The theorem doesn’t rule out any potential projective plane orders

q

≡

0

or

3

⁢

(

mod

⁢

4

)

,
but does rule out a large number of

q

≡

1

or

2

⁢

(

mod

⁢

4

)

, starting with

n

=

6

,

14

,

21

,

22

⁢

…

. Note in passing that it is easy to prove no prime
power falls foul of Bruck–Ryser, which is just as well, as (classical)
planes exist for all those orders.

The only other order ruled out to date is 10, via an epic computer search
by Lam, Swiercz and Thiel (read

http://www.cecm.sfu.ca/organics/papers/lam/index.htmlhttp://www.cecm.sfu.ca/organics/papers/lam/index.html

for Lam’s account).

All projective planes actually found to date, even the non-classical ones,
have an order

q

that is the power of a prime. The

prime power conjecture

says this is the case for all projective planes. It is still open.

References

1

BR49

R. H. Bruck

&

H. J. Ryser

,
“The non-existence of certain finite projective planes”

Can. J. Math.

12

(1949) pp 189–203

HR54

M. Hall, Jr.

&

H. J. Ryser

,
“

Normal completions

of incidence matrices”

Amer. J. Math.

76

(1954) pp 581–9

Hal59

Marshall Hall, Jr.

,

The Theory of Groups

,
Macmillan 1959

Cam94

Peter J. Cameron

,

Combinatorics

: topics, techniques, algorithms

,

Camb. Univ. Pr. 1994,

ISBN  0 521 45761 0

http://www.maths.qmul.ac.uk/ pjc/comb/http://www.maths.qmul.ac.uk/ pjc/comb/

(solutions, errata &c.)

CD96

Charles J. Colbourn

and

Jeffrey H. Dinitz

, eds.

The CRC Handbook of

Combinatorial Designs

,

CRC Press 1996,

ISBN  0 8493 8948 8

http://www.emba.uvm.edu/ dinitz/hcd.htmlhttp://www.emba.uvm.edu/ dinitz/hcd.html

(errata, new results)

the reference work on designs incl. Steiner systems,
proj. planes

dRe96

Marialuisa J. de Resmini

,
“Projective Planes, Nondesarguesian”,
pp 708–718 in the previous reference

vG03

Marijke van Gans

, M.Phil.(Qual.) thesis,
Birmingham 2003

Title

finite projective plane

Canonical name

FiniteProjectivePlane

Date of creation

2013-03-22 15:11:15

Last modified on

2013-03-22 15:11:15

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 51E15

Classification

msc 51A35

Classification

msc 05B25

Related topic

ProjectivePlane2

Related topic

IncidenceStructures

Related topic

Geometry

Related topic

DeBruijnErdHosTheorem

Defines

prime power conjecture

Defines

planar ternary ring

Defines

PTR

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