In my lecture today I proved that the Fano matroid is representable over a field F if and only the characteristic of F is 2. There is a proof of this using only the classical theorems of Ceva and Menelaus from Euclidean geometry. (This was not quite the proof I gave, since nobody knows these theorems now!) But I thought I would record it here.
Here is the Fano plane. As a matroid, the independent sets are all sets of at most two points, and the sets of three points which are not lines of the plane.
Consider the figure without the “curved” line. This is the diagram for Ceva’s theorem. It shows that, if we take x, y and z to be inhomogeneous coordinates of the points 6, 3 and 5 on the lines of the outer triangle, then xyz = +1, by Ceva’s Theorem.
Now consider the figure with only the outer lines of the triangle and the “curved” line, which is then a transversal to the three sides; so xyz = −1, by Menelaus’ Theorem.
So we must have +1 = −1, so that the characteristic is 2.