very symmetrical. Not so Euclidean geometry. In Euclidean geometry there are
parallel lines, that is pairs of lines which never meet. We can quite happily speak
of the concept of parallelism in Euclidean geometry. This is not possible in
projective geometry. In projective geometry all pairs of lines meet in a point. For
mathematicians this means Euclidean geometry is an inferior sort of geometry.
The Fano plane made Euclidean
If we remove one line and its points from the Fano plane we are once more
back in the realm of unsymmetrical Euclidean geometry and the existence of
parallel lines. Suppose we remove the ‘circular’ line DFG to give a Euclidean
diagram.
With one line fewer there are now six lines: AB, AC, AE, BC, BE and CE. There
are now pairs of lines which are ‘parallel’, namely AB and CE, AC and BE, and BC
and AE. Lines are parallel in this sense if they have no points in common – like
the lines AB and CE.
The Fano plane occupies an iconic position in mathematics because of its
connection to so many ideas and applications. It is one key to Thomas Kirkman’s