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%I #33 Oct 24 2021 02:02:04
%S 0,0,0,0,0,0,0,0,3,9,31,229
%N Number of geometrical configurations of type (n_3).
%C A geometrical configuration of type (n_3) consists of a set of n points in the Euclidean or extended Euclidean plane together with a set of n lines, such that each point belongs to 3 lines and each line contains 3 points.
%C Branko Grünbaum comments that it would be nice to settle the question as to whether all combinatorial configurations (13_3) are (as he hopes) geometrically realizable.
%D Many of the following references refer to combinatorial configurations (A001403) rather than geometrical configurations, but are included here in case they are helpful.
%D A. Betten and D. Betten, Regular linear spaces, Beitraege zur Algebra und Geometrie, 38 (1997), 111-124.
%D Bokowski and Sturmfels, Comput. Synthetic Geom., Lect Notes Math. 1355, p. 41.
%D CRC Handbook of Combinatorial Designs, 1996, p. 255.
%D D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination Chelsea, NY, 1952, Ch. 3.
%D F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
%D Pisanski, T. and Randic, M., Bridges between Geometry and Graph Theory, in Geometry at Work: Papers in Applied Geometry (Ed. C. A. Gorini), M.A.A., Washington, DC, pp. 174-194, 2000.
%D B. Polster, A Geometrical Picture Book, Springer, 1998, p. 28.
%D Sturmfels and White, Rational realizations..., in H. Crapo et al. editors, Symbolic Computation in Geometry, IMA preprint, Univ Minn., 1988.
%H A. Betten and D. Betten, <a href="https://doi.org/10.1007/BF01225670">Tactical decompositions and some configurations v_4</a>, J. Geom. 66 (1999), 27-41.
%H A. Betten, G. Brinkmann and T. Pisanski, <a href="https://doi.org/10.1016/S0166-218X(99)00143-2">Counting symmetric configurations v_3</a>, Discrete Appl. Math., 99 (2000), 331-338.
%H H. Gropp, <a href="http://dx.doi.org/10.1016/S0012-365X(96)00327-5">Configurations and their realization</a>, Discr. Math. 174 (1997), 137-151.
%H Jim Loy, <a href="https://web.archive.org/web/20140108030301/http://www.jimloy.com:80/geometry/desargue.htm">Desargues's Theorem</a>
%H Jim Loy, <a href="/A099999/a099999.gif">The configuration (10_3) arising from Desargues's theorem</a>
%H Tomo Pisanski, <a href="http://www.ijp.si/Configurations2004/papers.html">Papers on configurations</a>
%H T. Pisanski, M. Boben, D. Marušic, A. Orbanic and A. Graovac, <a href="https://doi.org/10.1016/S0012-365X(03)00110-9">The 10-cages and derived configurations</a>, Discrete Math. 275 (2004), 265-276.
%H B. Sturmfels and N. White, <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002038161">All 11_3 and 12_3 configurations are rational</a>, Aeq. Math., 39 1990 254-260.
%H Von Sterneck, <a href="https://doi.org/10.1007/BF01691614">Die Config. 11_3</a>, Monat. f. Math. Phys., 5 325-330 1894.
%H Von Sterneck, <a href="https://doi.org/10.1007/BF01696586">Die Config. 12_3</a>, Monat. f. Math. Phys., 6 223-255 1895.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Configuration.html">Configuration.</a>
%e The smallest examples occur for n = 9, where there are three configurations, one of which is the configuration arising from Pappus's Theorem (see the World of Mathematics "Configuration" link for drawings of all three).
%e The configuration arising from Desargues's theorem (see link above to an illustration) is one of the nine configurations for n = 10.
%Y Cf. A001403 (abstract or combinatorial configurations (n_3)), A023994, A100001, A098702, A098804, A098822, A098841, A098851, A098852, A098854.
%K nonn,nice,hard,more
%O 1,9
%A _N. J. A. Sloane_, following correspondence from Branko Grünbaum and Tomaz Pisanski, Nov 12 2004.
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