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 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.
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 schoolgirl problem .
In the theory of designing experiments the Fano plane appears as a protean example, a Steiner Triple System (STS). Given a finite number of n objects an STS is a way of dividing them into blocks of three so that every pair taken from the n objects is in exactly one block. Given the seven objects A, B, C, D, E, F and G the blocks in the STS correspond to the lines of the Fano plane.
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 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.
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.
In the theory of designing experiments the Fano plane appears as a protean example, a Steiner Triple System (STS). Given a finite number of n objects an STS is a way of dividing them into blocks of three so that every pair taken from the n objects is in exactly one block. Given the seven objects A, B, C, D, E, F and G the blocks in the STS correspond to the lines of the Fano plane.
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