Havliček-Tietze configurations in various projective planes

• Autorzy: Jakóbowski J. , Kacperek D.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Galois field; Havliček–Tietze configuration; homologic triangles; pascalian hexagon; projective plane; skew field of quaternions

PL

pola Galois; konfiguracja Havlíček-Tietze; trójkąt; sześciokąt; płaszczyzna rzutowa; kwaterniony

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 4
• Strony: 979--988
• Opis fizyczny: Bibliogr. 8 poz., rys.

Twórcy

autor

Jakóbowski J.

• Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, 10-719 Olsztyn, Poland

autor

Kacperek D.

• Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, 10-719 Olsztyn, Poland

Bibliografia

• [1] B. Grünbaum, Configurations of Points and Lines, AMS, Providence, Rhode Island, 2009.
• [2] P. Dembowski, Finite Geometries, Springer–Verlag, Berlin–Heidelberg–New York, 1968.
• [3] K. Havliček, J. Tietze, Zur Geometrie der endlichen Ebene der Ordnung n = 4, Czechoslovak Math. J. 21(96) (1971), 157–164.
• [4] T. Y. Lam, A First Course in Noncommutative Rings, Springer–Verlag, New York, Inc., 1991.
• [5] S. Lang, Algebra, Addison–Wesley Publishing Company, 1970.
• [6] A. Lewandowski, H. Makowiecka Some remarks on Havliček–Tietze configuration, Časopis pro pestováni matematiky 104 (1979), 180–184.
• [7] A. Lewandowski, H. Makowiecka A geometrical characterization of the projective plane of order 4, Časopis pro pĕstováni matematiky 104 (1979), 180–184.
• [8] C. R. Videla On the constructible numbers, Proc. Amer. Math. Soc. 127 (1999), 851–860.