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Configuration

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The word configuration is sometimes used to describe a finite collection of points p=(p_1,...,p_n), p_i in R^d, where R^d is a Euclidean space.

The term "configuration" also is used to describe a finite incidence structure (v_r,b_k) with the following properties, where v, b, r, and k are positive integers (Gropp 1997).

1. There are v points and b lines.

2. There are k points on each line and r lines through each point.

3. Two different lines intersect each other at most once and two different points are connected by a line at most once.

The conditions

vr=bk
(1)
v>=r(k-1)+1
(2)
b>=k(r-1)+1
(3)

are necessary for the existence of a configuration. For k=3, these conditions are also sufficient (Gropp 1994). For k=4, sufficiency was conjectured by Gropp (2001). The necessary conditions hold, but there is no 22_5. For k=6 and 7, they are not sufficient. Configurations with parameters (36_7,42_6) and (43_7,43_7) would be an affine plane and a projective plane of order 6, respectively, neither of which exists.

Configurations are among the oldest combinatorial structures, having been defined by T. Reye in 1876. An r-regular graph can be regarded as a configuration (v_r,b_2) by associating nodes with the points, and edges with the lines.

A symmetric configuration n_k=(n_k,n_k) consists of n lines and n points arranged such that k lines pass through each point and there are k points on each line. Important families are n3 configurations and n4 configurations. Betten et al. (2000) enumerated the abstract symmetric n3 configurations through n=18. The numbers of 7_3, 8_3, 9_3, ... configurations are 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ..., correcting an error of von Sterneck for 12_3 (OEIS A001403; Sterneck 1894, 1895; Wells 1991, p. 72; Colbourn and Dinitz 1996; Gropp 1997; Hilbert and Cohn-Vossen 1999).

A combinatorial configuration has a geometric realization when its abstract points and lines can be represented by points and straight lines with the prescribed incidences. Some authors take the ambient space to be the real projective plane, while others require an affine or Euclidean plane realization (Bokowski and Pilaud 2014, Berman et al. 2024).

The smallest geometric symmetric 4-configurations are the nonisomorphic Bokowski-Schewe configuration and Bokowski-Pilaud configuration; none exists for n<=17 (Bokowski and Schewe 2005, Bokowski and Schewe 2013, Bokowski and Pilaud 2014). No geometric 194 configuration exists (Bokowski and Pilaud 2015), while Grünbaum (2008) constructed the Grünbaum 204 configuration. At order 21, the Grünbaum-Rigby configuration and the Berman-Gévay-Pisanski configuration are the two geometric 21_4 polycyclic configurations (Berman et al. 2024).

FanoPlane

The Fano plane is the unique abstract 7_3 configuration. It is realizable over the finite field of order 2 GF(2), but not over the real or rational numbers (Gropp 1997). Allowing a point at infinity in the real projective plane does not remove this obstruction.

Moebius-KantorConfiguration

The unique abstract 8_3 configuration is the Möbius-Kantor configuration. It is realizable over the complex numbers, but not over the real or rational numbers (Gropp 1997). Thus it also has no realization by straight lines in the real projective plane.

Configurations9-3

Kantor (1891) showed that there are three 93 configurations, of which the Pappus configuration (left figure) is one (Coxeter 1950; Wells 1986, p. 75). The other two consist of embedded equilateral triangles (Wells 1991, pp. 159-160).

DesarguesConfiguration

Kantor (1881) proved that there are exactly ten 103 configurations, of which the Desargues configuration, illustrated above, is one. However, while not explicitly commented upon in paper, Kantor's (10_3)_4 contained a line which consists of two line segments oriented in slightly different directions. Schroeter (1889) subsequently proved that exactly one of these configurations cannot be drawn in the real or rational planes (Gropp 1997).

There are 31 configurations 11_3 (Gropp 1997), as constructed by Martinetti (1887) using recursive construction method, which were subsequently realized in the plane by Daublebsky von Sterneck (1895), though the proof that all are realizable came only with the work of Sturmfels and White (1990). Page and Dorwart (1984) discuss the 31 11_3 configurations (Wells 1991, p. 63).

There are 229 configurations 12_3, 228 of which had been found by Daublebsky von Sterneck (1895), with the missing one not found until the work of Gropp (1991). One of the 12_3 configurations is sometimes known as the Coxeter configuration, although it is perhaps better to refer it the Nauru configuration based on its Levi graph (which is the Nauru graph). Coordinates for realizations of all 11_3 and 12_3 configurations appeared in an appendix to Crapo et al. (1988).

The Coxeter configuration is a 28_3 configuration whose Levi graph is the Foster graph F_(056)C.

CremonaRichmondConfig

The Cremona-Richmond configuration, illustrated above, is one of the 245342 15_3 configurations.

There is a (2^(d-1))_d configuration known as the Cox configuration.

The following table summarizes the Levi graphs of some named configurations.

nameconfigurationLevi graph
Berman-Gévay-Pisanski configuration21_4Berman-Gévay-Pisanski graph
Bokowski-Pilaud configuration18_4Bokowski-Pilaud graph
Bokowski-Schewe configuration18_4Bokowski-Schewe graph
Cox configuration(2^(d-1))_dhypercube graph Q_d
Coxeter configuration28_3Foster graph F_(056)C
Cremona-Richmond configuration15_3Tutte 8-cage
Danzer configuration35_4Danzer graph
Desargues configuration10_3Desargues graph GP(10,3)
Fano plane7_3Heawood graph GH(1,2)
Georges configuration25_3Georges-Kelmans graph
Goedgebeur configuration15_3Goedgebeur graph
Gray configuration27_3Gray graph
Grünbaum 204 configuration20_4Grünbaum 204 graph
Grünbaum-Rigby configuration21_4Grünbaum-Rigby graph
Hesse configuration(9_412_3)Hesse graph
Klein configuration120-Klein graph
Kummer configuration16_6Kummer graph
Miquel configuration6_4rhombic dodecahedral graph
Möbius-Kantor configuration8_3Möbius-Kantor graph GP(8,3)
Nauru configuration12_3Nauru graph GP(12,5)
Pappus configuration9_3Pappus graph
Pasch configuration(6_24_3)Pasch graph
Reye configuration(12_416_3)Reye graph
Schläfli double six(30_212_5)Schläfli double six graph

Lu (2026) constructed geometric 234 configurations, settling the last unresolved existence case for geometric n4 configurations. Consequently, geometric n4 configurations in the real projective plane exist iff n>=18 and n!=19. This concerns geometric realizations, not merely abstract incidence structures.

See also

93 Configuration, 103 Configuration, 184 Configuration, 194 Configuration, 204 Configuration, 214 Configuration, 234 Configuration, Berman-Gévay-Pisanski Configuration, Bokowski-Pilaud Configuration, Bokowski-Schewe Configuration, Cox Configuration, Coxeter Configuration, Cremona-Richmond Configuration, Danzer Configuration, Desargues Configuration, Desmic Configuration, Double Sixes, Equilateral Triangle, Euclidean Space, Fano Plane, Framework, Georges Configuration, Graph Bar, Grünbaum 204 Configuration, Grünbaum-Rigby Configuration, Levi Graph, Möbius-Kantor Configuration, n3 Configuration, n4 Configuration, Orchard-Planting Problem, Ordinary Line Graph, Oriented Matroid, Pappus's Hexagon Theorem, Polycyclic Configuration, Projective Plane, Regular Graph, Reye Configuration, Rigid Graph, Tensegrity, Tesseract

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References

Berman, L. W.; Gévay, G.; and Pisanski, T. "On a New (21_4) Polycyclic Configuration." Elec. J. Combin. 31, P4.54, 2024. https://doi.org/10.37236/12405.Betten, A.; Brinkmann, G.; and Pisanski, T. "Counting Symmetric Configurations v_3." Disc. Appl. Math. 99, 331-338, 2000. https://doi.org/10.1016/S0166-218X(99)00143-2.Bokowski, J. and Pilaud, V. "Enumerating Topological (n_k)-Configurations." Comput. Geom. 47, 175-186, 2014. https://doi.org/10.1016/j.comgeo.2012.10.002.Bokowski, J. and Pilaud, V. "On Topological and Geometric (19_4) Configurations." Europ. J. Combin. 50, 4-17, 2015. https://doi.org/10.1016/j.ejc.2015.03.008.Bokowski, J. and Schewe, L. "There Are No Realizable 15_4- and 16_4-Configurations." Rev. Roumaine Math. Pures Appl. 50, 483-493, 2005.Bokowski, J. and Schewe, L. "On the Finite Set of Missing Geometric Configurations (n_4)." Comput. Geom. 46, 532-540, 2013. https://doi.org/10.1016/j.comgeo.2011.11.001.Bokowski, J. and Sturmfels, B. Computational Synthetic Geometry. Berlin, Germany: Springer-Verlag, p. 41, 1988.Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 255, 1996.Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.Crapo, H.; Havel, T.; Sturmfels, B.; Whiteley, W.; and White, N. "Symbolic Computations in Geometry." IMA Preprint No. 389. University of Minnesota, 1988.Daublebsky von Sterneck, R. "Die Configuration 11_3." Monatshefte f. Math. Phys. 5, 325-331, 1894.Daublebsky von Sterneck, R. "Die Configurationen 12_3." Monatshefte f. Math. Phys. 6, 223-255, 1895.Gropp, H. "Configurations and the Tutte Conjecture." Ars Combin. 29A, 171-177, 1990a.Gropp, H. "On the History of Configurations." In International Symposium on Structures in Mathematical Theories (Ed. A. Diez, J. Echeverria, and A. Ibarra). Bilbao, Spain, pp. 263-268, 1990b.Gropp, H. "Configurations and Steiner systems S(2,4,25) II--Trojan Configurations n_3." In Combinatorics '88, Research and Lecture Notes in Mathematics (Ed. A. Barlotti et al. ) Rende, Italy: Mediterranean Press, pp. 425-435, 1991.Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." Disc. Math. 101, 73-85, 1992a.Gropp, H. "Non-Symmetric Configurations with Deficiencies 1 and 2." Combinatorics '90. Recent Trends and Applications. Proceedings of the International Conference Held in Gaeta, May 20-27, 1990 (Ed. A. Barlotti, A. Bichera, P. V. Ceccherini, and G. Tallini). Amsterdam, Netherlands: North-Holland, pp. 227-239, 1992b.Gropp, H. "Nonsymmetric Configurations with Natural Index." Disc. Math. 124, 87-98, 1994. https://doi.org/10.1016/0012-365X(94)90087-6.Gropp, H. "Configurations and Their Realization." Disc. Math. 174, 137-151, 1997. https://doi.org/10.1016/S0012-365X(96)00327-5.Gropp, H. "Nonsymmetric Configurations with Line Size k=4." Elec. Notes Disc. Math. 8, 42-45, 2001. https://doi.org/10.1016/S1571-0653(05)80075-7.Grünbaum, B. "Musings on an Example of Danzer's." Europ. J. Combin. 29, 1910-1918, 2008. https://doi.org/10.1016/j.ejc.2008.01.004.Grünbaum, B. Configurations of Points and Lines. Providence, RI: Amer. Math. Soc., 2009.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, 1999.Kantor, S. "Die Configurationen (3,3)_10." Sitzungsber. Akad. Wiss. Wien Math.-Natur. Kl, 84, 1291-1314, 1881.Lu, W. "Maximally Symmetric (23_4) Configurations and the End of the Existence Problem for Geometric (n_4) Configurations." 8 Oct 2026. https://arxiv.org/abs/2610.14099.Martinetti, V. "Sulle configurazioni piane mu_3." Ann. Mat. Pura Appl. 15, 1-26, 1887.Page, W. and Dorwart, H. L. "Numerical Patterns and Geometrical Configurations." Math. Mag. 57, 82-92, 1984.Pisanski, T. and Randić, M. "Bridges between Geometry and Graph Theory." In Geometry at Work: A Collection of Papers Showing Applications of Geometry (Ed. C. A. Gorini). Washington, DC: Math. Assoc. Amer., pp. 174-194, 2000.Schroeter, H. "Üner die Bildungsweise und geometrische Konstruction der Konfigurationen 10_3.. Nachr. Ges. Wiss. Göttingen, 193-236, 1889.Sloane, N. J. A. Sequence A001403 in "The On-Line Encyclopedia of Integer Sequences."Sturmfels, B. and White, N. "All 11_3 and 12_3 Configurations are Rational." Aeq. Math. 39, 254-260, 1990.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 75, 1986.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 63 and 159-160, 1991.

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Configuration

Cite this as:

Weisstein, Eric W. "Configuration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Configuration.html

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