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Demonstratio Mathematica

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Volume 47, Issue 4


Havliček–Tietze Configurations in Various Projective Planes

Jan Jakóbowski / Danuta Kacperek
Published Online: 2014-12-11 | DOI: https://doi.org/10.2478/dema-2014-0078


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.

Keywords: and phrases Galois field; Havlicek-Tietze configuration; homologic triangles; pascalian hexagon; projective plane; skew field of quaternions


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About the article

Received: 2013-03-11

Published Online: 2014-12-11

Published in Print: 2014-12-01

Citation Information: Demonstratio Mathematica, Volume 47, Issue 4, Pages 979–988, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.2478/dema-2014-0078.

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© by Jan Jakóbowski. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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