Projective Configuration
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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, pro ...
, a configuration in the plane consists of a finite set of points, and a finite
arrangement of lines In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestr ...
, 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 Karl Theodor Reye (born 20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) was a German mathematician. He contributed to geometry, particularly projective geometry and synthetic geometry. He is best known for his ...
in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of
Desargues' theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...
.
Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
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, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
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. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s (these are said to be ''realizable'' in that geometry), or as a type of abstract
incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...
. In the latter case they are closely related to regular
hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) wh ...
s and biregular bipartite graphs, 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 In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we fo ...
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 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 them. The word is ...
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 points and lines of the Euclidean plane as the two types of objects and ignore al ...
s. For instance, there exist three different (93 93) configurations: the
Pappus configuration In geometry, the Pappus configuration is a Configuration (geometry), configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration i ...
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 Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
. 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 that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
of sides forms a configuration of type () * (43 62) and (62 43), the
complete quadrangle In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
and complete quadrilateral respectively. * (73), the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
. This configuration exists as an abstract
incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...
, but cannot be constructed in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
. * (83), the
Möbius–Kantor configuration In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of i ...
. 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 form ...
s. * (93), the
Pappus configuration In geometry, the Pappus configuration is a Configuration (geometry), configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration i ...
. * (94 123), the
Hesse configuration In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by , is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as t ...
of nine
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
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 P2(C), 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 \mathbf^3,\qquad (Z_1, ...
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 The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, ...
that shows that they cannot be given real-number coordinates. * (103), the
Desargues configuration In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues. The Desargues configuration can be constructed in two dimensions f ...
. * (124 163), the
Reye configuration In geometry, the Reye configuration, introduced by , is a configuration of 12 points and 16 lines. Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye ...
. * (125 302), the
Schläfli double six In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subse ...
, 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 th ...
* (153), the
Cremona–Richmond configuration In mathematics, the Cremona–Richmond configuration is a configuration of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles. It was studied by and . It is a generalized quadrangle wit ...
, formed by the 15 lines complementary to a double six and their 15 tangent planes * (166), the
Kummer configuration In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the points. Further, every pair of points is incident ...
. * (214), the Grünbaum–Rigby configuration. * (273), the Gray configuration * (354), Danzer's configuration. * (6015), the
Klein configuration In geometry, the Klein configuration, studied by , is a geometric configuration related to Kummer surface In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the ma ...
.


Duality of configurations

The projective dual 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 () configurations. Any
finite projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order is an (( configuration. Let be a projective plane of order . Remove from a point and all the lines of which pass through (but not the points which lie on those lines except for ) and remove a line not passing through and all the points that are on line . The result is a configuration of type ((. If, in this construction, the line is chosen to be a line which does pass through , then the construction results in a configuration of type ((. Since projective planes are known to exist for all orders 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.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 In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its ''order'', and the largest distance between two of its m ...
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 the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
. 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 In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two mutually inscribed tetrahedra: each vertex of one tetrahedron lies on a face plane of the other tetrahed ...
, 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 In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subse ...
, 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 In geometry, the Miquel configuration is a configuration of eight points and six circles in the Euclidean plane, with four points per circle and three circles through each point.. Its Levi graph is the Rhombic dodecahedral graph, the skeleton ...
.


See also

*
Perles configuration In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from ...
, a set of 9 points and 9 lines which do not all have equal numbers of incidences to each other


Notes


References

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External links

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