|
Solution 14: The Projective Plane
|
|
14. It is not difficult to show that the Fano plane is the smallest projective plane. We know that the Fano plane has order 2. Were there to be a smaller projective plane, it must have order 1. If there exists a projective plane of order 1, then each line is incidenct with 1 + 1 = 2 points and each point is incident with 2 lines. In addition, there exist 12 + 1 + 1 = 3 lines and 3 points. However this contradicts the axioms for a projective plane, because the fourth axiom states that there must be a set of four points such that no three are colinear. This implies that any projective plane must then have at least 4 points. Therefore, a projective plane of order 1 can not exist. Therefore, the smallest projective plane that exists is of order 2, and it follows that there are 7 points and lines. It is easy to see that the Fano plane satisfies this. Now we must show that the Fano plane is the unique plane of order 2. Use the following diagram for the Fano plane:
(Because the Fano plane is homogeneous, we could have labeled these points in any way.) Now for any projective plane of order 2, suppose that its point set P = {0, 1, 2, 3, 4, 5, 6}. We know that any point, say point 1, must be connected to all other points and that each line must contain 3 points. Thus, the lines containing point 1 can be labeled {1,2,4}, {1,3,0} and {1,6,5} WLOG. In addition, we know that point 2 must also be contained in 3 lines. We already know that it is contained in {1,2,4} and that it must still be joined to points 3, 0, 6 and 5. So, these lines will be {2,3,5} and {2,0,6} (Note: we could have made these lines {2,0,5} or {2,3,6} but these lines are essentially the same and you will see how even if one chooses those lines, the rest of our conclusions will still follow). So now we see that points 4, 0 and 5 do not lie on a line together so we must create a line containing these three points. In addition, we must create a line containing points 4, 3 and 6. We have created all 7 of our lines and this is the only way that we could have constructed this plane.
|
