View source for Configuration (geometry)

{{short description|Points and lines with equal incidences}}
{{about|points and lines|incidences of polytopes|Configuration (polytope)}}
[[File:Complete-quads.svg|thumb|upright=1.35|Configurations (4<sub>3</sub>6<sub>2</sub>) (a [[complete quadrangle]], at left) and (6<sub>2</sub>4<sub>3</sub>) (a complete quadrilateral, at right).]]
In [[mathematics]], specifically [[projective geometry]], a '''configuration''' in the plane consists of a finite set of [[Point (geometry)|points]], and a finite [[arrangement of lines]], such that each point is [[Incidence (geometry)|incident]] to the same number of lines and each line is incident to the same number of points.<ref>In the literature, the terms ''projective configuration'' {{harv|Hilbert|Cohn-Vossen|1952}} and ''tactical configuration of type (1,1)'' {{harv|Dembowski|1968}} are also used to describe configurations as defined here.</ref>

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 [[David Hilbert|Hilbert]] and [[Cohn-Vossen]]'s 1932 book ''Anschauliche Geometrie'', reprinted in English as {{harvtxt|Hilbert|Cohn-Vossen|1952}}.

Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the [[Euclidean plane|Euclidean]] or [[projective plane]]s (these are said to be ''realizable'' in that geometry), or as a type of abstract [[incidence geometry]]. In the latter case they are closely related to [[regular graph|regular]] [[hypergraph]]s and [[Biregular graph|biregular]] [[bipartite graph]]s, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the [[girth (graph theory)|girth]] of the corresponding bipartite graph (the [[Levi graph]] of the configuration) must be at least six.

==Notation==
A configuration in the plane is denoted by ({{math|''p''<sub>γ</sub> ''ℓ''<sub>{{pi}}</sub>}}), where {{math|''p''}} is the number of points, {{math|''ℓ''}} the number of lines, {{math|γ}} the number of lines per point, and {{pi}} the number of points per line. These numbers necessarily satisfy the equation

:<math>p\gamma = \ell\pi\,</math>

as this product is the number of point-line incidences (''flags'').

Configurations having the same symbol, say ({{math|''p''<sub>γ</sub> ''ℓ''<sub>{{pi}}</sub>}}), need not be [[Isomorphic (mathematics)|isomorphic]] as [[incidence structure]]s.  For instance, there exist three different (9<sub>3</sub>&nbsp;9<sub>3</sub>) configurations: the [[Pappus configuration]] and two less notable configurations.

In some configurations, {{math|1=''p'' = ''ℓ''}} and consequently, {{math|1=''γ'' = {{pi}}}}. These are called ''symmetric''  or ''balanced'' configurations{{sfn|Grünbaum|2009}} and the notation is often condensed to avoid repetition. For example, (9<sub>3</sub>&nbsp;9<sub>3</sub>) abbreviates to&nbsp;(9<sub>3</sub>).

==Examples==
[[File:Non-Desargues configuration.svg|upright=1.2|thumb|right|A (10<sub>3</sub>) configuration that is not incidence-isomorphic to a [[Desargues configuration]]]]

Notable projective configurations include the following:
* (1<sub>1</sub>), the simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial.
* (3<sub>2</sub>), the [[triangle]]. Each of its three sides meets two of its three vertices, and vice versa. More generally any [[polygon]] of {{math|''n''}} sides forms a configuration of type ({{math|''n''<sub>2</sub>}})
* (4<sub>3</sub> 6<sub>2</sub>), the [[complete quadrangle]]
* (6<sub>2</sub> 4<sub>3</sub>), the [[Pasch configuration]], which includes the [[complete quadrilateral]]
* (7<sub>3</sub>), the [[Fano plane]]. This configuration exists as an abstract [[incidence geometry]], but cannot be constructed in the [[Euclidean plane]].
* (8<sub>3</sub>), 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 number]]s.
* (9<sub>3</sub>), the [[Pappus configuration]]
* (9<sub>4</sub> 12<sub>3</sub>), the [[Hesse configuration]] of nine [[inflection point]]s 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.{{sfn|Kelly|1986}}
* (10<sub>3</sub>), the [[Desargues configuration]]
* (12<sub>4</sub> 16<sub>3</sub>), the [[Reye configuration]]
* (12<sub>5</sub> 30<sub>2</sub>), the [[Schläfli double six]], formed by 12 of the 27 lines on a [[cubic surface]]
* (15<sub>3</sub>), the [[Cremona–Richmond configuration]], formed by the 15 lines complementary to a double six and their 15 tangent planes
* (15<sub>4</sub> 20<sub>3</sub>), the [[Cayley–Salmon configuration]]
* (16<sub>6</sub>), the [[Kummer configuration]]
* (21<sub>4</sub>), the [[Grünbaum–Rigby configuration]]
* (27<sub>3</sub>), the [[Gray graph|Gray configuration]]
* (35<sub>4</sub>), [[Danzer's configuration]]<ref>{{harvnb|Grünbaum|2008}}, {{harvnb|Boben|Gévay|Pisanski|2015}}</ref>
* (60<sub>15</sub>), the [[Klein configuration]]

==Duality of configurations==

The [[projective dual]] of a configuration ({{math|''p''<sub>γ</sub> ''ℓ''<sub>{{pi}}</sub>}}) is a ({{math|''ℓ''<sub>{{pi}}</sub> ''p''<sub>γ</sub>}}) configuration in which the roles of "point" and "line" are exchanged.  Types of configurations therefore come in dual pairs, except when taking the dual results in an isomorphic configuration.  These exceptions are called ''self-dual'' configurations and in such cases {{math|1=''p'' = ''ℓ''}}.<ref>{{harvnb|Coxeter|1999|loc=pp. 106–149}}</ref>

==The number of ({{math|''n''<sub>3</sub>}}) configurations==

The number of nonisomorphic configurations of type ({{math|''n''<sub>3</sub>}}), starting at {{math|1=''n'' = 7}}, is given by the sequence

:[[1 (number)|1]], [[1 (number)|1]], [[3 (number)|3]], [[10 (number)|10]], [[31 (number)|31]], [[229 (number)|229]], 2036, 21399, 245342, ... {{OEIS|id=A001403}}

These numbers count configurations as abstract incidence structures, regardless of realizability.{{sfn|Betten|Brinkmann|Pisanski|2000}}
As {{harvtxt|Gropp|1997}} discusses, nine of the ten (10<sub>3</sub>) configurations, and all of the (11<sub>3</sub>) and (12<sub>3</sub>) configurations, are realizable in the Euclidean plane, but for each {{math|''n'' ≥ 16}} there is at least one nonrealizable ({{math|''n''<sub>3</sub>}}) configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (12<sub>3</sub>) configurations, and found 228 of them, but the 229th configuration, the Gropp 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 ({{math|''p''<sub>γ</sub>}}) configurations.

Any [[Projective plane#Finite projective planes|finite projective plane]] of order {{math|''n''}} is an (({{math|''n''<sup>2</sup> + ''n'' + 1)<sub>''n'' + 1</sub>)}} configuration. Let {{math|Π}} be a projective plane of order {{math|''n''}}. Remove from {{math|Π}} a point {{math|''P''}} and all the lines of {{math|Π}} which pass through {{math|''P''}} (but not the points which lie on those lines except for {{math|''P''}}) and remove a line {{math|''ℓ''}} not passing through {{math|''P''}} and all the points that are on line {{math|''ℓ''}}. The result is a configuration of type (({{math|''n''<sup>2</sup> – 1)<sub>''n''</sub>)}}. If, in this construction, the line {{math|''ℓ''}} is chosen to be a line which does pass through {{math|''P''}}, then the construction results in a configuration of type (({{math|''n''<sup>2</sup>)<sub>''n''</sub>)}}. Since projective planes are known to exist for all orders {{math|''n''}} which are powers of primes, these constructions provide infinite families of symmetric configurations.

Not all configurations are realizable, for instance, a (43<sub>7</sub>) configuration does not exist.<ref>This configuration would be a projective plane of order 6 which does not exist by the [[Bruck–Ryser theorem]].</ref> However, {{harvtxt|Gropp|1990}} has provided a construction which shows that for {{math|''k'' ≥ 3}}, a ({{math|''p''<sub>k</sub>}}) configuration exists for all {{math|''p'' ≥ 2 ''ℓ''<sub>''k''</sub> + 1}}, where {{math|''ℓ''<sub>''k''</sub>}} is the length of an optimal [[Golomb ruler]] of order {{math|''k''}}.
==Unconventional configurations==
===Higher dimensions===

[[File:Double six.svg|thumb|upright=1.5|The [[Schläfli double six]].]]
The concept of a configuration may be generalized to higher dimensions,{{sfn|Gévay|2014}} for instance to points and lines or planes in [[Projective space|space]]. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane.

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 graph|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.

===Topological configurations===
Configuration in the projective plane that is realized by points and [[Arrangement_of_lines#Non-Euclidean_line_arrangements|pseudolines]] is called
topological configuration.{{sfn|Grünbaum|2009}} For instance, it is known that there exists no point-line (19<sub>4</sub>) configurations, however, there exists a topological configuration with these parameters.
===Configurations of points and circles===
Another generalization of the concept of a configuration concerns configurations of points and circles, a notable example being the (8<sub>3</sub> 6<sub>4</sub>) [[Miquel configuration]].{{sfn|Grünbaum|2009}}

==See also==
*[[Perles configuration]], a set of 9 points and 9 lines which do not all have equal numbers of incidences to each other

==Notes==
{{reflist}}

==References==
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==External links==
{{Commons category|Configurations (geometry)}}
*{{mathworld | urlname = Configuration | title = Configuration|mode=cs2}}

[[Category:Configurations (geometry)| ]]