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Havliček-Tietze configurations in various projective planes

Jakóbowski J., Kacperek D.

Demonstratio Mathematica

2014

Vol. 47, nr 4

979--988

A. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havliček–Tietze configuration (shortly H - T) in the desarguesian projective plane is equivalent to existence in the associated field, a root of polynomial x2 + x + 1, different from 1. We show that such a configuration exists in every projective plane over Galois field GF(p2) for p ≠ 3. As it has been demonstrated, in a projective plane over arbitrary field F, each hexagon contained in H - T, satisfies the Pappus–Pascal axiom, even if F is noncommutative. Moreover, such a hexagon either is pascalian or has exactly one pair of opposite sides intersecting at a point collinear with two points not belonging to these sides. In particular, all such hexagons are pascalian iff char F = 2. For the (noncommutative) field of quaternions, we have determined the set of all roots of the mentioned polynomial. Every H - T is the special Pappus configuration, in which three main diagonals of the hexagon are concurrent.