14

Finite Geometries, Codes,

Latin Squares,

and Other Pretty Creatures

14.1 Small Exotic Worlds

The Fano plane is a really small world (Figure 14.1). It only has 7 points,
which form 7 lines. In the figure, 6 of these lines are straight and one
is circular; but for the inhabitants of this tiny world (the Fanoans), the
straight and curved lines look the same. Also, in our figure it seems that the
circular line intersects some of the straight lines twice; but the intersection
points that are not marked are not real intersection points, they are there
only because we want to draw a picture of this very different world in our
Euclidean plane.

FIGURE 14.1. The Fano plane.

The Fanoans are very proud of their world. They say that it is tiny but
perfect in various ways. They point out that

(a)through any two points their world has a unique line.