The Pascal mysticum demystified
Pascal's famous theorem states that opposite edges in a hexagon inscribed into a conic intersect in three collinear points. The 60 hexagons with the same vertices but different edges give rise to a total of 60 such ``Pascal lines''. This article gives a nice summary of developments in the 19th century starting from this observation. The 60 Pascal lines meet in threes in 20 ``Steiner points'' that lie in fours on 15 ``Plücker lines''. The Pascal lines also meet in threes in 60 ``Kirkman points'' that lie in threes on 20 ``Cayley lines''. Finally, the Cayley lines meet in fours at 15 ``Salmon nodes''. These incidences in this configuration of 95 points and 95 lines are presented in a highly mnemonic diagram and proved in a systematic and clear way. The first ingredient in the proofs is a permutation notation that labels the various points and lines by elements of the symmetric group \(S_6\). The second ingredient is Desargues' theorem which allows to deduce collinearity of points from concurrency of lines and vice versa. The proofs are than a matter of inferring the sought incidences from properly labelled Desargues configurations. The most important contribution of this paper is the very neat permutation notation. It allows to circumvent the combinatorial complexity in formulation and derivation of theorems.
scientific article; zbMATH DE number 6130591