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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
, a configuration in the plane consists of a finite set of
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
, 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 Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...
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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...
and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
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
hypergraph In mathematics, a hypergraph is a generalization of a Graph (discrete mathematics), graph in which an graph theory, edge can join any number of vertex (graph theory), vertices. In contrast, in an ordinary graph, an edge connects exactly two vert ...
s and biregular
bipartite graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...
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 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 (), where is the number of points, the number of lines, the number of lines per point, and 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 (''flags''). Configurations having the same symbol, say (), need not be
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
as
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the Point (geometry), points and Line (geometry), lines of the Euclidean plane as t ...
s. For instance, there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations. In some configurations, and consequently, . These are called ''symmetric'' or ''balanced'' configurations and 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. Often excluded as being trivial. * (32), the
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
. Each of its three sides meets two of its three vertices, and vice versa. More generally any
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
of sides forms a configuration of type () * (43 62), the complete quadrangle * (62 43), the
Pasch configuration In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four Point (geometry), points in a Plane (geometry), plane, no three of which are ...
, which includes the complete quadrilateral * (73), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
. * (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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. * (93), the Pappus configuration * (94 123), the Hesse configuration of nine
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...
s of a
cubic curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...
in the
complex projective plane In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...
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. * (103), the Desargues configuration * (124 163), the Reye configuration * (125 302), the Schläfli double six, formed by 12 of the 27 lines on a
cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than ...
* (153), the Cremona–Richmond configuration, formed by the 15 lines complementary to a double six and their 15 tangent planes * (154 203), the Cayley–Salmon configuration * (166), the Kummer configuration * (214), the Grünbaum–Rigby configuration * (273), the Gray configuration * (354), Danzer's configuration * (6015), the Klein configuration


Duality of configurations

The
projective dual In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one t ...
of a configuration () is a () 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 .


The number of () configurations

The number of nonisomorphic configurations of type (), starting at , 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 discusses, nine of the ten (103) configurations, and all of the (113) and (123) configurations, are realizable in the Euclidean plane, but for each there is at least one nonrealizable () 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, 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 (''p''γ) configurations.


Cyclic configurations

Some self dual configurations (''p''''k'') are cyclic configurations and can be constructed by one "generator line", like , with vertices indexed from zero, and where indices in following lines are cycled forward modulo ''p''. This is guaranteed to produce a symmetric configuration when valid. An invalid generator line produces disconnected configurations, or it may break the axiom requiring at most one line between any two points. Every
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
as configuration (''p''2) is trivially a cyclic configuration with generator line . A triangle (32) has lines . The Fano plane, (73), the smallest self-dual order 3 symmetric configuration, can be defined by generator line as lines . They also can be represented in configuration table: The smallest self-dual order 5 symmetric configuration is (215) is a cyclic configuration and can be generated by the line .


Finite projective planes

Any
finite projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
of order ''n'', PG(2,''n''), is an ''n''2+''n''+1)''n''+1configuration. Since projective planes are known to exist for all orders ''n'' which are powers of primes, these constructions provide infinite families of symmetric configurations.
Automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s for PG(2,''n''), with ''n''=''q''''m'' (''q'' prime) are (''m''!)(''n''3-1)(''n''3-''n'')(''n''3-''n''2)/(''n''-1). Not all symmetric configurations are realizable. Specifically ''n'' must be a power prime. For instance, PG(2,6) or (437) configuration does not exist.This configuration would be a projective plane of order 6 which does not exist by the Bruck–Ryser theorem. However, has provided a construction which shows that for , a () configuration exists for all , where is the length of an optimal Golomb ruler of order .


Unconventional configurations


Higher dimensions

The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
. 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 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 pseudolines is called topological configuration. For instance, it is known that there exists no point-line (194) 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 (83 64) Miquel configuration.


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


References

*. *. *. * * *. * *. *. *. *. *. *. * .


External links

*{{mathworld , urlname = Configuration , title = Configuration, mode=cs2