Havliček–Tietze Configurations in Various Projective Planes

Jan Jakóbowski 1  and Danuta Kacperek 1
  • 1 FACULTY OF MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF WARMIA AND MAZURY IN OLSZTYN 10-719 OLSZTYN, POLAND

Abstract

A. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havlicek-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 charF=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.

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