%I
%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 (A001405) 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), 111124.
%D A. Betten and D. Betten, Tactical decompositions and some configurations v_4, J. Geom. 66 (1999), 2741.
%D A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331338.
%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. CohnVossen, Geometry and the Imagination Chelsea, NY, 1952, Ch. 3.
%D F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
%D Pisanski, T.; Boben, M.; Marusic, D.; Orbanic, A.; and Graovac, A. The 10cages and derived configurations. Discrete Math. 275 (2004), 265276.
%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. 174194, 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.
%D Sturmfels and White, All 11_3 and 12_3 configurations are rational, Aeq. Math., 39 1990 254260.
%D Von Sterneck, Die Config. 11_3, Monat. f. Math. Phys. 5 325330 1894; Die Config. 12_3, op. cit. 6 223255 1895.
%H H. Gropp, <a href="http://dx.doi.org/10.1016/S0012365X(96)003275">Configurations and their realization</a>, Discr. Math. 174 (1997), 137151.
%H Jim Loy, <a href="http://www.jimloy.com/math/math.htm">Mathematics Page</a> (see Desargues's Theorem)
%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 B. Sturmfels and N. White, <a href="http://resolver.sub.unigoettingen.de/purl?GDZPPN002038161">All 11_3 and 12_3 configurations are rational</a>, Aeq. Math., 39 1990 254260.
%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 familiar configuration arising from Desargues's theorem (see illustration).
%Y Cf. A001405 (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.
