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 Anschaulische 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, or as abstract incidence structures. In the latter case they are closely related to regular hypergraphs and regular bipartite graphs.

Notation

$pgamma\; =\; lpi,$

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.

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

Examples

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.
• (10₃), the Desargues configuration.
• (15₃), the Cremona-Richmond configuration.

Duality of configurations

The projective dual to a configuration (p_γ l_π) is a configuration (l_π p_γ). Thus, types of configurations come in dual pairs, except when p = l and the configuration type is self-dual.

The number of (n₃ n₃) configurations

The number of nonisomorphic configurations of type (n₃n₃), starting at n = 7, is given by the sequence

1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ...

These numbers count configurations as abstract incidence structures, regardless of realizability. As Gropp (1997) discusses, nine of the ten (10₃10₃) configurations, and all of the (11₃11₃) and (12₃12₃) configurations, are realizable in the Euclidean plane, but for each n ≥ 16 there is at least one nonrealizable (n₃n₃) configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (12₃12₃) configurations, and found 228 of them, but the 229th configuration was not discovered until 1988.

Higher dimensions

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 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 45 points, 27 lines, three lines per point, and five points per line.

Coxeter (1948) further generalises a configuration in 3 dimensions to points, lines and planes, and generally in n dimensions to all j-dimensional elements (for 0 >= j > n).

References

• Berman, Leah W. "Movable (n₄) configurations". The Electronic Journal of Combinatorics 13 (1): R104. See also Berman's animations of movable configurations
• Coxeter, H.S.M. (1948). Regular Polytopes, Methuen and Co.
• Gropp, Harald (1997). "Configurations and their realization". Discrete Mathematics 174 (1–3): 137–151.
• Grünbaum, Branko "Configurations of points and lines". The Coxeter Legacy: Reflections and Projections, 179–225. .

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