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A099999 Number of geometrical configurations of type (n_3). 3
0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 31, 229 (list; graph; refs; listen; history; text; internal format)



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.


Many 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.

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

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

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.


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

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

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

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

Tomo Pisanski, Papers on configurations

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

Eric Weisstein's World of Mathematics, Configuration.


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: A255382 A089475 A299549 * A039749 A034538 A034540

Adjacent sequences:  A099996 A099997 A099998 * A100000 A100001 A100002




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



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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)