Configuration (geometry)

Configurations (4₃6₂) (a complete quadrangle, at left) and (6₂4₃) (a complete quadrilateral, at right).

In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.

Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie (reprinted in English as Geometry and the Imagination).

Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as abstract incidence structures. In the latter case they are closely related to regular hypergraphs and regular bipartite graphs.

Notation

A configuration in the plane is denoted by (p_γ ℓ_π), where p is the number of points, ℓ the number of lines, γ the number of lines per point, and π the number of points per line. These numbers necessarily satisfy the equation

$p\gamma = \ell\pi\,$

as this product is the number of point-line incidences.

The notation (p_γ ℓ_π) does not determine a projective configuration up to incidence isomorphism. For instance, there exist three different (9₃ 9₃) configurations: the Pappus configuration and two less notable configurations.

In some configurations, p = ℓ and γ = π. These are called symmetric configurations and the notation is often condensed to avoid repetition. For example (9₃ 9₃) abbreviates to (9₃).

Examples

A (10₃) configuration that is not incidence-isomorphic to a Desargues configuration

Notable projective configurations include the following:

• (1₁), the simplest possible configuration, consisting of a point incident to a line.
• (3₂), the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally any polygon of n sides forms a configuration of type (n₂)
• (4₃ 6₂) and (6₂ 4₃), the complete quadrangle and complete quadrilateral respectively.
• (7₃), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the Euclidean plane.
• (8₃), the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers.
• (9₃), the Pappus configuration.
• (9₄ 12₃), the configuration of nine inflection points of a cubic curve in the complex projective plane and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane the property that it contains every line through its points; configurations with this property are known as Sylvester–Gallai configurations due to the Sylvester–Gallai theorem that shows that they cannot be given real-number coordinates (Kelly 1986).
• (10₃), the Desargues configuration.
• (15₃), the Cremona–Richmond configuration.
• (12₄ 16₃), the Reye configuration.

Duality of configurations

The projective dual to a configuration (p_γ l_π) is a configuration (l_π p_γ) in which the roles of "point" and "line" are exchanged. Thus, types of configurations come in dual pairs, except when taking the dual results in an isomorphic configuration. Such configurations are self-dual and in such cases p = l.

The number of (n₃) configurations

The number of nonisomorphic configurations of type (n₃), starting at n = 7, is given by the sequence

1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ... (sequence A001403 in OEIS)

These numbers count configurations as abstract incidence structures, regardless of realizability (Betten, Brinkmann & Pisanski 2000). As Gropp (1997) discusses, nine of the ten (10₃) configurations, and all of the (11₃) and (12₃) configurations, are realizable in the Euclidean plane, but for each n ≥ 16 there is at least one nonrealizable (n₃) configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (12₃) configurations, and found 228 of them, but the 229th configuration was not discovered until 1988.

Constructions of symmetric configurations

There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric (p_γ) configurations.

Any finite projective plane of order n is an (n² + n + 1)_{n + 1} configuration. Let π be a projective plane of order n. Remove from π a point P and all the lines of π which pass through P (but not the points which lie on those lines except for P) and remove a line l not passing through P and all the points that are on line l. The result is a configuration of type (n² - 1)_n. If, in this construction, the line l is chosen to be a line which does pass through P, then the construction results in a configuration of type (n²)_n. Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.

Not all configurations are realizable, for instance, a (43₇) configuration does not exist. However, Gropp (1990) has provided a construction which shows that for k ≥ 3, a (p_k) configuration exists for all p ≥ 2 l_k + 1, where l_k is the length of an optimal Golomb ruler of order k.

Higher dimensions

The Schläfli double six.

The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in space. Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahedra, Reye's configuration, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, and the Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line.

A further generalization is obtained in three dimensions by considering incidences of points, lines and planes, or j-spaces (0 ≤ j < 3), where each j-space is incident with N_jk k-spaces (j ≠ k). Writing $N_{jj}$ for the number of j-spaces present. a given configuration may be represented by the matrix:

$\begin{vmatrix}\begin{vmatrix}N_{00} & N_{01} & N_{02} \\ N_{10} & N_{11} & N_{12} \\ N_{20} & N_{21} & N_{22}\end{vmatrix}\end{vmatrix}$

The principle extends generally to n dimensions, where 0 ≤ j < n. Such configurations are related mathematically to regular polytopes.

References

• Berman, Leah W., "Movable (n₄) configurations", The Electronic Journal of Combinatorics 13 (1): R104, http://www.combinatorics.org/Volume_13/Abstracts/v13i1r104.html . See also Berman's animations of movable configurations.
• Betten, A; Brinkmann, G.; Pisanski, T. (2000), "Counting symmetric configurations", Discrete Applied Mathematics 99 (1–3): 331–338, doi:10.1016/S0166-218X(99)00143-2 .
• Coxeter, H.S.M. (1948), Regular Polytopes, Methuen and Co .
• Coxeter, H.S.M. (1999), "Self-dual configurations and regular graphs", The Beauty of Geometry, Dover, ISBN 0-486-40919-8 
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3540617868, MR0233275 
• Gropp, Harald (1990), "On the existence and non-existence of configurations n_k", Journal of Combinatorics and Information System Science 15: 34-48 
• Gropp, Harald (1997), "Configurations and their realization", Discrete Mathematics 174 (1–3): 137–151, doi:10.1016/S0012-365X(96)00327-5 .
• Grünbaum, Branko (2006), "Configurations of points and lines", in Davis, Chandler; Ellers, Erich W., The Coxeter Legacy: Reflections and Projections, American Mathematical Society, pp. 179–225 .
• Grünbaum, Branko (2009), Configurations of Points and Lines, Graduate Studies in Mathematics, 103, American Mathematical Society, ISBN 978-0-8218-4308-6 .
• Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), Chelsea, pp. 94–170, ISBN 0-8284-1087-9 .
• Kelly, L. M. (1986), "A resolution of the Sylvester–Gallai problem of J. P. Serre", Discrete and Computational Geometry 1 (1): 101–104, doi:10.1007/BF02187687 .

External links

• Weisstein, Eric W., "Configuration" from MathWorld.