Triangles in Euclidean Arrangements
The number of triangles in arrangements of lines and pseudolines has been Object of some research. Most results, however, concern arrangements in the projective plane. We obtain results for the number of triangles in Euclidean arrangements of pseudolines. Though the Change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs.
In 1926 Levi proved that a nontrivial arrangement – simple or not – of n pseudolines in the projective plane contains n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains n – 2 triangles, we had to find a completely different proof. On the other hand a non-simple arrangements of n pseudolines in the Euclidean plane tan have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and some extensions.
Mathematics Subject Classifications (1991) 52A10, 52ClO.
KeywordsArrangement Euclidean plane pseudoline strechability triangle
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- 2.Füredi, Z., Palasti, I.: Arrangements of lines with large number of triangles. In: Proc. Am. Math. Soc., vol. 92, pp. 561–566 (1984)Google Scholar
- 4.Grünbaum, B.: Arrangements and spreads. Regional Conf. Ser. Math., Amer. Math. Soc., Providence, RI (1972)Google Scholar
- 5.Harborth, H.: Some simple arrangements of pseudolines with a maximum number of triangles, Discrete geometry and convexity. In: Proc. Conf., vol. 440, New York (1982); Ann. N. Y. Acad. Sci., 31-33 (1985)Google Scholar
- 8.Ringel, G.: Teilungen der Ebenen durch Geraden oder topologische Geraden. Math. Z. 64, 79–102 (1956)Google Scholar
- 10.Roudneff, J.-P.: Arrangements of lines with a minimum number of triangles are simple. Discrete Comput. Geom. 3, 97–102 (1988)Google Scholar
- 11.Roudneff, J.-P.: The maximum number of triangles in arrangements of pseudolines. J. Comb. Theory, Ser. B 66, 44–74 (1996)Google Scholar