finite projective plane
Finite projective planes from axioms
Let us start from the beginning and assume we are given a set of things called points and a set of things called lines, 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 points, there is exactly one line through both of them.
For any two distinct lines, there is exactly one point on both of them.
Given these http://planetmath.org/node/6940projective plane axioms we can apply the De Bruijn–Erdős theorem twice (once with the rôle of points and lines swapped) which tells us we must have , and one of two possible configurations:
The other solution is a projective plane:
the number of points and the number of lines are of the form for some integer ;
each point is incident with lines and each line with points.
One usually avoids the degenerate case by demanding there is a quadrangle in the plane: four points 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 and . 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.
Projective planes as bipartite graphs
Projective planes are, apart from anything else, also bipartite graphs: let the points be represented by “black” nodes (vertices) of the graph, and let the lines also be nodes, “white” ones say. Edges will represent incidence between a “black” and a “white” node, so there are 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 points there is a path via the line they share, and for any two lines 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 PQ would mean two lines intersecting in two different points, and vice versa. So the girth (minimum cycle length) is 6 (showing there indeed are 6-cycles for 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.
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 of different symbols for coordinate values, two of which are 0 and 1. Start with four points O, I, X, Y forming a quadrangle. XY is the “line at infinity” .
Construction: O and I are given coordinates and , and the other points on OI (except its intersection with ) arbitrarily with unique . For any point P not on , let YP intersect OI at , and let XP intersect OI at . Then P will be given . For the coordinates of points on , intersect the lines joining with , for all , with , calling the intersection . Finally, Y gets coordinate (instead of in the literature).
Lines not through Y are given coordinates where and are their intersections with and with OY respectively; lines through Y (other than ) get coordinates going by their intersection with OX. The line 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 say.
There are more points (other than Y) incident to , let’s call them Y where ranges over the different values of the set . Again, each Y is incident to more lines (apart from ), let’s call those where is another subscript from . Likewise there are more lines (other than ) incident to Y, let’s call them and to each of them are incident more points (apart from Y) which we can label Y.
This immediately produces the incidence list of the preceding section. Pictorially, for (with the “to be assessed” portion filled in as thinner lines):
We already have containing all points Y, and in addition if we wanted we could easily arrange for to be incident with all the Y (just renumber the labels in the sets Y for every ), and for to be incident with all the Y (just look at which Y are incident with , then exercise the freedom we still have to relabel all the , matching each time the corresponding 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, along the top we fan out in two stages to and items, along the bottom we also fan out in two stages to and items, and the hard part is the right. We have by now accounted for edges so we know there must be exactly more edges, between some of those points and some of those lines. But where?
From the projective plane properties it is easy to see that any point Y must, for every value of , be incident to no more than one of the (because if it was incident to two they would intersect both in our Y and in their shared Y at the bottom). On the other hand, Y needs more mates so it needs to choose exactly one for every value of .
By exactly the same reasoning, any needs to be incident with, for every value of , at most one and then exactly one Y. In other words, the projective plane structure takes the form of, for each pair of values and , a bijection between the indices and the indices.
This is not yet the only constraint. A closer analysis [vG03] shows that, in order to prevent any two lines meeting in more than one point or any two points in more than two lines, all compound bijections of the form
(for and ) 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.
Planar ternary rings
We’ve been giving our points and lines cooordinates from a set of different labels. We can give the structure of a ternary ring, that is, a set with a ternary operation , by defining
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- designs, or what’s the same thing, Steiner systems . This way, we already assume in the definition that each “block” (line) contains exactly points, from which it follows by calculation that the number of lines that contain a given point is also , and that the total number of lines equals the total number of points.
Also, the unique incidence property that every two points lie together on exactly one line is given, and the dual one that every two lines 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 points on a line and the number of lines 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 . We know there are points and the same number of lines. Let be an 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 is known as the incidence matrix of the plane.
has rows and columns both representing lines, and the entries here are the dot products of any two rows of , i.e. the number of points any two lines and have in common, which is when and 1 otherwise, by the projective plane properties.
where is the identity matrix (ones along the diagonal, zeros everywhere else) and is the matrix filled with ones in every entry. Note has this same value for every projective plane with points and lines, while 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, having this value is the only thing needed to make the plane specified by satisfy the projective plane properties; it is a fully equivalent formulation of those properties.
(once) with eigenvector for which
by letting the matrix act on these eigenvectors. This gives
which in our case () amounts to
As is even and the other factor a square, we get an integer
where the sign is moot, as has no correspondence between its columns and rows.
The Bruck–Ryser theorem [BR49] states that
If is the order of a projective plane, and or , then is the sum of two integer squares (one of which can be ).
It can be proven [BR49, Cam94] using the incidence matrix , by a clever rearrangement of (plus one extra term) for an arbitrary vector , 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 , hence the constraint on (mod 4).
Interestingly, there is a partial converse [HR54]: whenever 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 matrix (where ) such that . 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 or , but does rule out a large number of or , starting with . 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 that is the power of a prime. The prime power conjecture says this is the case for all projective planes. It is still open.
- 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
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.)
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|
|Date of creation||2013-03-22 15:11:15|
|Last modified on||2013-03-22 15:11:15|
|Last modified by||Mathprof (13753)|
|Defines||prime power conjecture|
|Defines||planar ternary ring|