Introduction to finite geometry II

Given an affine plane, we can introduce points at “infinity” and join those points by a line to get a system in which there are no parallel lines and any two distinct lines determine a unique point! More formally, what we do is this :

  1. Corresponding to each parallel class we take a single new point and add that point to all the lines of that class. Different parallel classes get different points and these points are called points at infinity.
  2. All the new points at infinity (and only these) form a single new line called the line at infinity.

Note here that infinity is just a terminology, origins of which will be clear if we study the history of projective geometry.

Projective Plane

Def : A projective plane is a triple {(P,L,I)} where {P} is a set of points, {L} a set of lines and {I \subseteq P \times L} an incidence relation such that :

  1. Each line is incident with at least three points and each point is incident with at least three lines.
  2. Any two distinct points determine (/ lie on) a unique line.
  3. Any two distinct lines determine (/ intersect at) a unique point.

Note that the definition of a projective plane has a duality between lines and points; if the words lines and points are exchanged in the definition, then all the axioms are preserved. Thus any proof about lines can be turned into a proof about points, and vice versa.

Theorem 1 Every affine plane can be extended to a projective plane and every projective plane has an affine plane imbedded in it.

Proof: One part of theorem, affine planes being extended to projective planes, is clear from the discussion at the beginning. Now, given a projective plane, pick a line and delete it along with all the points incident to it. It can be easily checked that the system you get is an affine plane. \Box

We denote the projective plane that we get from {AG(2,F)} as {PG(2,F)}. From above theorem, it is clear that we can talk about order of finite projective planes as well.

Theorem 2 Let {(P,L,I)} be a finite projective plane, then there exists a number {n \geq 2} called the order of the plane such that :

  1. Each line is incident with {n+1} points.
  2. Each point is incident with {n+1} lines.
  3. There are {n^2 + n + 1} lines.
  4. There are {n^2 + n + 1} points.

Proof: Exercise! \Box

Both affine planes and projective planes are specific examples of a more general structure called an Incidence system.

Def : An incidence system is a triple {(X,B,I)} where {X} is a set of points, {B} a set of blocks and {I \subseteq X \times B} is a binary relation called the incidence relation.

Def : A {2-(v,k,\lambda)} design is an incidence system with {v} points such that each block is incident with {k} points and any {2} distinct points are incident with exactly {\lambda} blocks.

Examples :

  1. An affine plane of order {n} is a {2-(n^2, n, 1)} design.
  2. A projective plane of order {n} is a {2-(n^2 + n + 1, n + 1, 1)} design.

Theorem 3 Given a {2-(v,k,\lambda)} design, there are constants {r,b} such that each point is incident with exactly {r} blocks and there are total {b} blocks. These are given by

\displaystyle r(k-1) = \lambda (v-1)

and

\displaystyle bk = rv

.

Proof: Let {a} be a point. Let {r_a} be the number of blocks {a} is incident to. Count the number of ordered pairs {(x,B)} where {x} is a point different from {a} and {B} is a block to which both {x} and {a} are incident. First choosing {x} and then {B} we get {(v-1)\lambda} such pairs. First choosing {B}, then {x} we get {r_a(k-1)} such pairs. Therefore, {r_a = \frac{\lambda (v-1)}{k-1}}. But the right hand side is a constant, independent of the choice of {a}. Therefore we get a constant {r} given by {r = \frac{\lambda (v-1)}{k-1}} such that each point is incident to exactly {r} blocks.
Now count the number of ordered pairs {(B,x)} where {B} is a block incident to the point {x}. {B} can be chosen in {b} ways and then it has {k} points to which it is incident. Or, we can choose a point first in {v} ways and then there are {r} blocks incident to it. Therefore {bk = rv}. \Box

Theorem 4 A {2-(n^2, n, 1)} design is an affine plane.

Proof: Only the playfair axiom need to be checked as rest all is pretty direct from the definition of a design. Let {x} be a point and {l} a line not incident to it. Since we have {r = (n^2 - 1)/(n-1) = n + 1}, there are {n+1} lines through {x}. Since {x} doesn’t lie on {l}, each of the {n} points on {l} determine a unique line with {x}. So we are only left with {1} more line passing through {x}. This must be disjoint with {l} and hence parallel to it. \Box

Theorem 5 A {2-(n^2 + n + 1, n + 1, 1)} design is a Projective plane.

Proof: We just need to check that two lines (blocks) intersect at a unique point. It suffices to show that no two lines are disjoint since if they do intersect then the point must be unique. Let {l} be a line and {p} a point not incident to it. There are {n+1} points incident to {l} and {\frac{n^2 + n}{n} = n + 1} lines passing through {p}. Each point on {l} determines a unique line along with {p}. Therefore no line through {p} can be parallel (disjoint) to {l}. Hence no two lines can be disjoint. \Box

Therefore, finite affine planes are precisely {2-(n^2, n, 1)} designs and finite projective planes are precisely {2-(n^2 + n + 1, n + 1, 1)} designs.

Can we characterize the integer values of {(v,k, \lambda)} for which there is a {2-(v,k,\lambda)} design? Theorem 3 above does give some necessary conditions that these parameters must satisfy but full characterization is a fundamental and still unresolved problem in design theory. For example, the values {v = 22}, {k = 7}, {\lambda = 2}, {r = 7}, {b = 22} satisfy {bk = rv} and {r(k-1) = \lambda(v-1)} but it can be shown that there is no {2-(22, 7, 2)} design. Also, we do not know yet if there is a {2-(22, 8, 4)} design or not!

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About Anurag Bishnoi

Currently a maths PhD student at Ghent University working in the Incidence Geometry research group. I am broadly interested in combinatorics, finite geometry and group theory.
This entry was posted in Finite Geometry. Bookmark the permalink.

8 Responses to Introduction to finite geometry II

  1. Anirban Mandal says:

    A very logical presentation, but you should put in your own commentaries and write in a more colloquail (or flippant if you wish) style, make it both serious and fun to read, as if you yourself in person is teaching someone. And is it Tex typeset? It has been very beautifully done.

    • Hmm, thanks for the comment. I was focusing more on the content and hoping that it’d speak for itself. But yes, I can try what you have suggested. I’ll try it in my next post, a recurring “trick” in combinatorics, double counting.

    • And yes, I am using latex. You can see the links I have given on the post “hello”

  2. Signify says:

    grt job!! i tried to understand the concept of an affine plane but then going to this is too much. So i would restrain myself from going any furthur :P….but the job is commendable. The expressions you are putting, the formatting…everything is marked with an assiduous effort. So best of luck.. and yes as the previous commentator stressed….the tutorials can be made a little more like feynman…though its too much to ask…but still it would make it more lucid.

  3. 🙂 I promise the next post would be quite easy to understand. Hopefully, I ll post it by tonight.

  4. Pingback: Count Twice! « Anurag's Math Blog

  5. Pingback: Point-Line Geometries | Anurag's Math Blog

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