A099999 Number of geometrical configurations of type (n_3).

0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 31, 229

Offset

1,9

Comments

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.

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.

References

Many of the following references refer to combinatorial configurations (A001403) 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.

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

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

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

F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.

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.

Links

Table of n, a(n) for n=1..12.

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.

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

Jim Loy, Desargues's Theorem

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

Tomo Pisanski, Papers on configurations

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

B. Sturmfels and N. 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.

Von Sterneck, Die Config. 12_3, Monat. f. Math. Phys., 6 223-255 1895.

Eric Weisstein's World of Mathematics, Configuration.

Example

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).
The configuration arising from Desargues's theorem (see link above to an illustration) is one of the nine configurations for n = 10.

Crossrefs

Cf. A001403 (abstract or combinatorial configurations (n_3)), A023994, A100001, A098702, A098804, A098822, A098841, A098851, A098852, A098854.

Sequence in context: A089475 A351185 A299549 * A039749 A034538 A034540

Adjacent sequences: A099996 A099997 A099998 * A100000 A100001 A100002

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane, following correspondence from Branko Grünbaum and Tomaz Pisanski, Nov 12 2004.

Status

approved