The Fano plane is the projective plane over the field $\mathbf Z/2$. It can be used to remember octonion multiplication, as nicely explianed in John Baez's article on octonions (see http://math.ucr.edu/home/baez/octonions/).
The picture (taken from Baez's website) is as follows:
It indicates for example, using the cyclic orderings on the lines, that $e_6 \cdot e_1= e_5$ but that $e_6 \cdot e_4= -e_3$.
Two natural questions arise for me:
Why did Baez label the circles in such a weird order. There is probably a good reason which is implicit. (A priori, if I permute the labels arbitrarily, I'll get something isomorphic).
Also, one could decide to choose other orientations for the arrows. So again, why did Baez choose these orientations? And what could we get with other orientations?