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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, incidence geometry is the study of
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. A geometric structure such as the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidence structure'' is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries. Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects. In
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
they are called
hypergraph In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

hypergraph
s, and in
combinatorial design theory Combinatorial design theory is the part of combinatorics, combinatorial mathematics that deals with the existence, construction and properties of set system, systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and ...
they are called
block design In combinatorics, combinatorial mathematics, a block design is an incidence structure consisting of a set together with a Family of sets, family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain condition ...
s. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics. In the examples selected for this article we use only those with a natural geometric flavor. A special case that has generated much interest deals with finite sets of points in the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and what can be said about the number and types of (straight) lines they determine. Some results of this situation can extend to more general settings since only incidence properties are considered.


Incidence structures

An ''incidence structure'' consists of a set whose elements are called ''points'', a disjoint set whose elements are called ''lines'' and an ''incidence relation'' between them, that is, a subset of whose elements are called ''flags''. If is a flag, we say that is ''incident with'' or that is incident with (the terminology is symmetric), and write . Intuitively, a point and line are in this relation if and only if the point is ''on'' the line. Given a point and a line which do not form a flag, that is, the point is not on the line, the pair is called an ''anti-flag''.


Distance in an incidence structure

There is no natural concept of distance (a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geomet ...
) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest
path A path is a route for physical travel - see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire pat ...
between two vertices in this
bipartite graph In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

bipartite graph
. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again uses a graph-theoretic notion in a related structure, this time the ''collinearity graph'' of the incidence structure. The vertices of the collinearity graph are the points of the incidence structure and two points are joined if there exists a line incident with both points. The distance between two points of the incidence structure can then be defined as their distance in the collinearity graph. When distance is considered in an incidence structure, it is necessary to mention how it is being defined.


Partial linear spaces

Incidence structures that are most studied are those that satisfy some additional properties (axioms), such as
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, affine planes,
generalized polygon In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, partial geometries and near polygons. Very general incidence structures can be obtained by imposing "mild" conditions, such as: A
partial linear space A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space (geometry), linear space. The notion is equivalent to that of a ...
is an incidence structure for which the following axioms are true: * Every pair of distinct points determines at most one line. * Every line contains at least two distinct points. In a partial linear space it is also true that every pair of distinct lines meet in at most one point. This statement does not have to be assumed as it is readily proved from axiom one above. Further constraints are provided by the regularity conditions: RLk: Each line is incident with the same number of points. If finite this number is often denoted by . RPr: Each point is incident with the same number of lines. If finite this number is often denoted by . The second axiom of a partial linear space implies that . Neither regularity condition implies the other, so it has to be assumed that . A finite partial linear space satisfying both regularity conditions with is called a ''tactical configuration''. Some authors refer to these simply as '' configurations'', or ''projective configurations''. If a tactical configuration has points and lines, then, by double counting the flags, the relationship is established. A common notation refers to -''configurations''. In the special case where (and hence, ) the notation is often simply written as . A ''linear space'' is a partial linear space such that: * Every pair of distinct points determines exactly one line. Some authors add a "non-degeneracy" (or "non-triviality") axiom to the definition of a (partial) linear space, such as: * There exist at least two distinct lines. This is used to rule out some very small examples (mainly when the sets or have fewer than two elements) that would normally be exceptions to general statements made about the incidence structures. An alternative to adding the axiom is to refer to incidence structures that do not satisfy the axiom as being ''trivial'' and those that do as ''non-trivial''. Each non-trivial linear space contains at least three points and three lines, so the simplest non-trivial linear space that can exist is a triangle. A linear space having at least three points on every line is a Sylvester–Gallai design.


Fundamental geometric examples

Some of the basic concepts and terminology arises from geometric examples, particularly
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and affine planes.


Projective planes

A ''projective plane'' is a linear space in which: * Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition: * There exist four points, no three of which are
collinear In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
. There is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
between and in a projective plane. If is a finite set, the projective plane is referred to as a ''finite'' projective plane. The order of a finite projective plane is , that is, one less than the number of points on a line. All known projective planes have orders that are
prime power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. A projective plane of order is an configuration. The smallest projective plane has order two and is known as the ''Fano plane''.


Fano plane

This famous incidence geometry was developed by the Italian mathematician
Gino Fano Gino Fano (5 January 18718 November 1952) was an Italians, Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contr ...

Gino Fano
. In his work on proving the independence of the set of axioms for projective ''n''-space that he developed, he produced a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line had only three points on it. The planes in this space consisted of seven points and seven lines and are now known as
Fano plane In finite geometry, the Fano plane (after Gino Fano) is the Projective plane#Finite projective planes, finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 l ...

Fano plane
s. The Fano plane cannot be represented in the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
using only points and straight line segments (i.e., it is not realizable). This is a consequence of 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, ...
, according to which every realizable incidence geometry must include an ''ordinary line'', a line containing only two points. The Fano plane has no such line (that is, it is a
Sylvester–Gallai configuration In geometry, a Sylvester–Gallai configuration consists of a finite subset of the points of a projective space with the property that the line through any two of the points in the subset also passes through at least one other point of the subset. ...
), so it is not realizable. A
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 (geometry), plane, no three of which are on a common line, and ...
consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the ''Fano axiom'', often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear.


Affine planes

An ''affine plane'' is a linear space satisfying: * For any point and line not incident with it (an anti-flag) there is exactly one line incident with (that is, ), that does not meet (known as
Playfair's axiom In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): ''In a plane (mathematics), plane, given a line and a point not on it, at most one line parallel (geometry), parallel to ...
), and satisfying the non-degeneracy condition: * There exists a triangle, i.e. three non-collinear points. The lines and in the statement of Playfair's axiom are said to be ''parallel''. Every affine plane can be uniquely extended to a projective plane. The ''order'' of a finite affine plane is , the number of points on a line. An affine plane of order is an configuration.


Hesse configuration

The affine plane of order three is a configuration. When embedded in some ambient space it is called the
Hesse configuration In geometry, the Hesse configuration, introduced by Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish Scottish usually refers to something of, from, or related to Scotland, inclu ...

Hesse configuration
. It is not realizable in the Euclidean plane but is realizable in the
complex projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
as the nine
inflection point In differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

inflection point
s of an
elliptic curve In mathematics, an elliptic curve is a Nonsingular variety, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point ''O''. An elliptic curve is defined over a field ...
with the 12 lines incident with triples of these. The 12 lines can be partitioned into four classes of three lines apiece where, in each class the lines are mutually disjoint. These classes are called ''parallel classes'' of lines. Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any single line and all the points on that line produces this affine plane of order three (it is also unique). Removing one point and the four lines that pass through that point (but not the other points on them) produces the
Möbius–Kantor configuration In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
.


Partial geometries

Given an integer , a tactical configuration satisfying: * For every anti-flag there are flags such that and , is called a ''partial geometry''. If there are points on a line and lines through a point, the notation for a partial geometry is . If these partial geometries are
generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 ...
s. If these are called
Steiner system image:Fano plane.svg, 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In Combinatorics, combinatorial mathematics, a Steiner s ...
s.


Generalized polygons

For , a generalized -gon is a partial linear space whose incidence graph has the property: * The
girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth (g ...
of (length of the shortest
cycle Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...
) is twice the
diameter In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
of (the largest distance between two vertices, in this case). A ''generalized 2-gon'' is an incidence structure, which is not a partial linear space, consisting of at least two points and two lines with every point being incident with every line. The incidence graph of a generalized 2-gon is a complete bipartite graph. A generalized -gon contains no
ordinary -gon
ordinary -gon
for and for every pair of objects (two points, two lines or a point and a line) there is an ordinary -gon that contains them both. Generalized 3-gons are projective planes. Generalized 4-gons are called
generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 ...
s. By the Feit-Higman theorem the only finite generalized -gons with at least three points per line and three lines per point have = 2, 3, 4, 6 or 8.


Near polygons

For a non-negative integer a near -gon is an incidence structure such that: * The maximum distance (as measured in the collinearity graph) between two points is , and * For every point and line there is a unique point on that is closest to . A near 0-gon is a point, while a near 2-gon is a line. The collinearity graph of a near 2-gon is a
complete graph In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

complete graph
. A near 4-gon is a generalized quadrangle (possibly degenerate). Every finite generalized polygon except the projective planes is a near polygon. Any connected bipartite graph is a near polygon and any near polygon with precisely two points per line is a connected bipartite graph. Also, all dual polar spaces are near polygons. Many near polygons are related to finite simple groups like the
Mathieu group In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
s and the
Janko group J2 In the area of modern algebra known as group theory, the Janko group ''J2'' or the Hall-Janko group ''HJ'' is a sporadic simple group of Order (group theory), order :   2733527 = 604800 : ≈ 6. History and properties ''J2'' is ...
. Moreover, the generalized 2''d''-gons, which are related to
Groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
, are special cases of near 2''d''-gons.


Möbius planes

An abstract Mōbius plane (or inversive plane) is an incidence structure where, to avoid possible confusion with the terminology of the classical case, the lines are referred to as ''cycles'' or ''blocks''. Specifically, a Möbius plane is an incidence structure of points and cycles such that: * Every triple of distinct points is incident with precisely one cycle. * For any flag and any point not incident with there is a unique cycle with and . (The cycles are said to ''touch'' at .) * Every cycle has at least three points and there exists at least one cycle. The incidence structure obtained at any point of a Möbius plane by taking as points all the points other than and as lines only those cycles that contain (with removed), is an affine plane. This structure is called the ''residual'' at in design theory. A finite Möbius plane of ''order'' is a tactical configuration with points per cycle that is a 3-design, specifically a block design.


Incidence theorems in the Euclidean plane


The Sylvester-Gallai theorem

A question raised by J.J. Sylvester in 1893 and finally settled by
Tibor Gallai Tibor Gallai (born Tibor Grünwald, 15 July 1912 – 2 January 1992) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic groups ...
concerned incidences of a finite set of points in the Euclidean plane. Theorem (Sylvester-Gallai): A finite set of points in the Euclidean plane is either
collinear In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
or there exists a line incident with exactly two of the points. A line containing exactly two of the points is called an ''ordinary line'' in this context. Sylvester was probably led to the question while pondering about the embeddability of the Hesse configuration.


The de Bruijn–Erdős theorem

A related result is the de Bruijn–Erdős theorem.
Nicolaas Govert de Bruijn Nicolaas Govert (Dick) de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of mathematical analysis, analysis, number theory, combinatorics and logic.Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician In this page we keep the names in Hungarian order (family name first). {{compact ToC , short1, side=yes A * Alexits György (1899–1 ...

Paul Erdős
proved the result in the more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is: ::In a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, every non-collinear set of points determines at least distinct lines. As the authors pointed out, since their proof was combinatorial, the result holds in a larger setting, in fact in any incidence geometry in which there is a unique line through every pair of distinct points. They also mention that the Euclidean plane version can be proved from the Sylvester-Gallai theorem using
induction Induction may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the t ...
.


The Szemerédi–Trotter theorem

A bound on the number of flags determined by a finite set of points and the lines they determine is given by: Theorem (Szemerédi–Trotter): given points and lines in the plane, the number of flags (incident point-line pairs) is: :O \left ( n^ m^ + n + m \right ), and this bound cannot be improved, except in terms of the implicit constants. This result can be used to prove Beck's theorem. A similar bound for the number of incidences is conjectured for point-circle incidences, but only weaker upper bounds are known.


Beck's theorem

Beck's theorem says that finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points. The theorem asserts the existence of positive constants such that given any points in the plane, at least one of the following statements is true: # There is a line that contains at least of the points. # There exist at least lines, each of which contains at least two of the points. In Beck's original argument, is 100 and is an unspecified constant; it is not known what the optimal values of and are.


More examples

* Projective geometries *
Moufang polygon In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible building (mathematics), buildings of rank two that admit the action of root groups. In a book on the topic, Tit ...
* Schläfli double six *
Reye configuration In mathematics, the Reye configuration, introduced by , is a Configuration (geometry), 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 o ...

Reye configuration
*
Cremona–Richmond configuration In mathematics, the Cremona–Richmond configuration is a configuration (geometry), 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 ...
* Kummer configuration *
Klein configuration In geometry, the Klein configuration, studied by , is a geometric configuration 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, ...
*
Non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective sp ...
s


See also

*
Combinatorial design Combinatorial design theory is the part of combinatorics, combinatorial mathematics that deals with the existence, construction and properties of set system, systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and ...
s *
Finite geometry A finite geometry is any geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that a ...
* Intersection theorem *
Levi graph In combinatorics, 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 configu ...


Notes


References

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


External links

*
''incidence system''
at the
Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate ...
{{Incidence structures