I'm attempting to solve a little exercise that my maths lecturer has given me relating to projective planes. Basically, she has asked me to show that a projective plane contains a set of 7 points and 6 lines. I was wondering how this can be done, as using the axioms that define a projective plane, I can only achieve a Fano plane which contains 7 points and 7 lines as a circular line has to be included to satisfy the axioms.

The axioms in question are:

  1. Any two distinct points lie on a unique line.
  2. Any two distinct lines meet in a unique point.
  3. There exist at least three non-collinear points.
  4. There exist at least three points on every line.

Any tips or help will be greatly appreciated!

  • 1
    $\begingroup$ aHave you considered the possibility that your lecturer may be wrong? Could there be a typo in the exercise? $\endgroup$ – bof Jun 3 '18 at 19:22
  • $\begingroup$ I wanted to know whether it was possible to solve it for six lines and seven points before asking whether the exercise contained a typo or error. But I'll admit that the thought had crossed my mind. $\endgroup$ – Maths Matador Jun 3 '18 at 19:26

The exercise is wrong; any projective plane has as many lines as points. Let me try to give you the simplest proof from the axioms that $7$ points and $6$ lines is impossible.

There are $\binom72=21$ (unordered) pairs of points. Since each pair of points determines a line, if there are fewer then $7$ lines, then some line must be determined by more than $3$ pairs of points; i.e., some line must contain more than $3$ points. It will suffice to show that, if there are more than $3$ points on some line, then there are more than $7$ points all told.

Suppose that there are $4$ distinct points $Q_1, Q_2, Q_3, Q_4$ on the line $L.$ By Axiom 3, there is a point $P$ which is not on the line $L.$ Let $L_i$ be the line through $P$ and $Q_i.$ It's easy to see that the lines $L,L_1,L_2,L_3,L_4$ are all distinct. Each line $L_i$ must contain another point, call it $R_i,$ besides $P$ and $Q_i.$ Then the nine points $P,Q_1,Q_2,Q_3,Q_4,R_1,R_2,R_3,R_4$ are all distinct.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.