The Fano planeThe Fano plane
The Fano plane geometry was discovered at about the same time as Pick’s
formula, but has nothing to do with measuring anything at all. Named after the
Italian mathematician Gino Fano, who pioneered the study of finite geometry, the
Fano plane is the simplest example of a ‘projective’ geometry. It has only seven
points and seven lines.
The seven points are labelled A, B, C, D, E, F and G. It is easy to pick out six
of the seven lines but where is the seventh? The properties of the geometry and
the way the diagram is constructed make it necessary to treat the seventh line as
DFG – the circle passing through D, F and G. This is no problem since lines in
discrete geometry do not have to be ‘straight’ in the conventional sense.
This little geometry has many properties, for example:
- every pair of points determines one line passing through both,
- every pair of lines determines one point lying on both.
These two properties illustrate the remarkable duality which occurs in
geometries of this kind. The second property is just the first with the words
‘point’ and ‘line’ swapped over, and likewise the first is just the second with the
same swaps.
If, in any true statement, we swap the two words and make small adjustments
to correct the language, we get another true statement. Projective geometry is