1,9

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.

Barnko Gruenbaum comments that it would be nice to settle the question as to whether all combinatorial configurations (13_3) are (as he hopes) geometrically realizable.

Most of the following references refer to combinatorial configurations (A001405) rather than geometrical configurations, but are included here in case they are helpful.

A. Betten and D. Betten, Regular linear spaces, Beitraege zur Algebra und Geometrie, 38 (1997), 111-124.

A. Betten and D. Betten, Tactical decompositions and some configurations v_4, J. Geom. 66 (1999), 27-41.

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.

Bokowski and Sturmfels, Comput. Synthetic Geom., Lect Notes Math. 1355, p. 41.

CRC Handbook of Combinatorial Designs, 1996, p. 255.

H. Gropp, Configurations and their realization, Discr. Math. 174 (1997), 137-151.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination Chelsea, NY, 1952, Ch. 3.

Pisanski, T.; Boben, M.; Marusic, D.; Orbanic, A.; and Graovac, A. The 10-cages and derived configurations. Discrete Math. 275 (2004), 265-276.

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.

B. Polster, A Geometrical Picture Book, Springer, 1998, p. 28.

Sturmfels and White, Rational realizations..., in H. Crapo et al. editors, Symbolic Computation in Geometry, IMA preprint, Univ Minn., 1988.

Sturmfels and White, All 11_3 and 12_3 configurations are rational, Aeq. Math., 39 1990 254-260.

Von Sterneck, Die Config. 11_3, Monat. f. Math. Phys. 5 325-330 1894; Die Config. 12_3, op. cit. 6 223-255 1895.

Jim Loy, Mathematics Page (see Desargues's Theorem)

Jim Loy, The configuration (10_3) arising from Desargues's theorem

Tomo Pisanski, Papers on configurations

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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).

Cf. A001405 (abstract or combinatorial configurations (n_3)), A023994, A100001, A098702, A098804, A098822, A098841, A098851, A098852, A098854.

Sequence in context: A027040 A111063 A089475 this_sequence A039749 A034538 A034540

Adjacent sequences: A099996 A099997 A099998 this_sequence A100000 A100001 A100002

nonn,nice,hard,more

N. J. A. Sloane (njas(AT)research.att.com), following correspondence from Branko Gruenbaum and Tomaz Pisanski, Nov 12 2004.