Abstract
Try for yourself — before you read much further — to construct configurations of points in the plane that determine “relatively few” slopes. For this we assume, of course, that the n ≥ 3 points do not all lie on one line. Recall from Chapter 8 on “Lines in the plane” the theorem of Erdos and de Bruijn: the n points will determine at least n different lines. But of course many of these lines may be parallel, and thus determine the same slope.
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References
J. E. Goodman and R. Pollack: A combinatorial perspective on some problems in geometry, Congressus Numerantium 32 (1981), 383–394.
R. E. Jamison and D. Hill: A catalogue of slope-critical configurations,Congressus Numerantium 40 (1983), 101–125.
P. R. Scott: On the sets of directions determined by n points, Amer. Math. Monthly 77 (1970), 502–505.
P. Ungar: 2N noncollinear points determine at least 2N directions, J. Combinatorial Theory Ser. A 33 (1982), 343–347.
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© 1998 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (1998). The slope problem. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22343-7_9
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DOI: https://doi.org/10.1007/978-3-662-22343-7_9
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