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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.

Central automorphisms of Veblenian nearaffine planes

Cudna-Lasecka K., Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 3

337-347

The paper deals with nearaffine planes described by H. A. Wilbrink. We consider their central automorphisms, i.e. automorphisms satisfying the Veblen condition, which become central collineations in connected projective planes. Moreover, a concept of central pseudo-automorphism is considered, i.e. some bijections in a nearaffine plane are not automorphisms but they become central collineations in the related projective planes.