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In [[mathematics]], '''incidence geometry''' is the study of [[incidence structure]]s. A geometric structure<!--Do not wikilink, the redirect is inappropriate--> such as the [[Euclidean plane]] is a complicated object that involves concepts such as length, angles, continuity, betweenness, and [[incidence (geometry)|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.<ref>As, for example, L. Storme does in his chapter on Finite Geometry in {{harvtxt|Colbourn|Dinitz|2007|loc=pg. 702}}</ref>

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]] they are called [[hypergraph]]s, and in [[Combinatorial design|combinatorial design theory]] they are called [[block design]]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]] 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 ==
{{main|incidence structure}}

An ''incidence structure'' {{math|(''P'', ''L'', I)}} consists of a set {{math|''P''}} whose elements are called ''points'', a disjoint set {{math|''L''}} whose elements are called ''lines'' and an ''incidence relation'' {{math|I}} between them, that is, a subset of {{math|''P'' × ''L''}} whose elements are called ''flags''.<ref>Technically this is a rank two incidence structure, where rank refers to the number of types of objects under consideration (here, points and lines). Higher ranked structures are also studied, but several authors limit themselves to the rank two case, and we shall do so here.</ref> If {{math|(''A'', ''l'')}} is a flag, we say that {{math|''A''}} is ''incident with'' {{math|''l''}} or that {{math|''l''}} is incident with {{math|''A''}} (the relation is symmetric), and write {{math|''A'' I ''l''}}. Intuitively, a point and line are in this relation if and only if the point is ''on'' the line. Given a point {{math|''B''}} and a line {{math|''m''}} which do not form a flag, that is, the point is not on the line, the pair {{math|(''B'', ''m'')}} is called an ''anti-flag''.

===Distance in an incidence structure===
There is no natural concept of distance (a [[Metric (mathematics)|metric]]) in an incidence structure. However, a combinatorial metric does exist in the corresponding [[Levi graph|incidence graph (Levi graph)]], namely the length of the shortest [[Path (graph theory)|path]] between two vertices in this [[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]]s, [[Affine plane (incidence geometry)|affine planes]], [[generalized polygon]]s, [[partial geometry|partial geometries]] and [[near polygon]]s. Very general incidence structures can be obtained by imposing "mild" conditions, such as:

A [[partial linear space]] is an incidence structure for which the following axioms are true:<ref>{{harvnb|Moorhouse|loc=pg.5}}</ref>
* 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 {{math|''k''}}.

'''RPr''': Each point is incident with the same number of lines. If finite this number is often denoted by {{math|''r''}}.

The second axiom of a partial linear space implies that {{math|''k'' > 1}}. Neither regularity condition implies the other, so it has to be assumed that {{math|''r'' > 1}}.

A finite partial linear space satisfying both regularity conditions with {{math|''k'', ''r'' > 1}} is called a ''tactical configuration''.<ref>{{harvnb|Dembowski|1968|page=5}}</ref> Some authors refer to these simply as ''[[configuration (geometry)|configurations]]'',<ref>{{citation
| last=Coxeter
| first=H. S. M.
| author-link=H.S.M. Coxeter
| title=Introduction to Geometry
| location=New York
| publisher=John Wiley & Sons
| year=1969
| page=233
| isbn=978-0-471-50458-0}}</ref> or ''projective configurations''.<ref>{{citation
 | last1 = Hilbert | first1 = David | author1-link = David Hilbert
 | last2 = Cohn-Vossen | first2 = Stephan | author2-link = Stephan Cohn-Vossen
 | edition = 2nd
 | isbn = 978-0-8284-1087-8
 | pages = 94–170
 | publisher = Chelsea
 | title = Geometry and the Imagination
 | year = 1952}}</ref> If a tactical configuration has {{math|''n''}} points and {{math|''m''}} lines, then, by double counting the flags, the relationship {{math|1=''nr'' = ''mk''}} is established. A common notation refers to {{math|(''n''<sub>''r''</sub>, ''m''<sub>''k''</sub>)}}-''configurations''. In the special case where {{math|1=''n'' = ''m''}} (and hence, {{math|1=''r'' = ''k''}}) the notation {{math|(''n''<sub>''k''</sub>, ''n''<sub>''k''</sub>)}} is often simply written as {{math|(''n''<sub>''k''</sub>)}}.

[[File:Sample Incidence.jpg|thumb|Simplest non-trivial linear space]]

A [[Linear space (geometry)|''linear space'']] is a partial linear space such that:<ref>{{harvnb|Moorhouse|loc=pg. 5}}</ref>
* 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.<ref>There are several alternatives for this "non-triviality" axiom. This could be replaced by "there exist three points not on the same line" as is done in {{harvtxt|Batten|Beutelspacher|1993|loc=pg. 1}}. There are other choices, but they must always be ''existence'' statements that rule out the very simple cases to exclude.</ref>

This is used to rule out some very small examples (mainly when the sets {{math|''P''}} or {{math|''L''}} 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 configuration|Sylvester–Gallai design]].

== Fundamental geometric examples ==
Some of the basic concepts and terminology arises from geometric examples, particularly [[projective plane]]s and [[Affine plane (incidence geometry)|affine planes]].

===Projective planes===
{{main|projective plane}}
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]].

There is a [[bijection]] between {{math|''P''}} and {{math|''L''}} in a projective plane. If {{math|''P''}} is a finite set, the projective plane is referred to as a ''finite'' projective plane. The '''order''' of a finite projective plane is {{math|1=''n'' = ''k'' – 1}}, that is, one less than the number of points on a line. All known projective planes have orders that are [[prime power]]s. A projective plane of order {{math|''n''}} is an {{math|((''n''<sup>2</sup> + ''n'' + 1)<sub>''n'' + 1</sub>)}} configuration.

The smallest projective plane has order two and is known as the ''Fano plane''.
[[File:Fano plane.svg|thumb|<center>Projective plane of order 2 <br> the Fano plane</center>]]

==== Fano plane ====
{{main|Fano plane}}

This famous incidence geometry was developed by the Italian mathematician [[Gino Fano]]. In his work<ref>{{citation|first=G.|last=Fano|title=Sui postulati fondamentali della geometria proiettiva|year=1892|journal=Giornale di Matematiche|volume= 30|pages=106–132}}</ref> on proving the independence of the set of axioms for [[Projective space|projective ''n''-space]] that he developed,<ref>{{harvnb|Collino|Conte|Verra|2013|loc=p. 6}}</ref> 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.<ref>{{harvnb|Malkevitch}} Finite Geometries? an AMS Featured Column</ref> The planes in this space consisted of seven points and seven lines and are now known as [[Fano plane]]s.

The Fano plane cannot be represented in the [[Euclidean plane]] using only points and straight line segments (i.e., it is not realizable). This is a consequence of the [[Sylvester–Gallai theorem]], 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]]), so it is not realizable.{{sfnp|Aigner|Ziegler|2010}}

A [[complete quadrangle]] 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===

{{main|affine plane (incidence geometry)}}

An ''affine plane'' is a linear space satisfying:
* For any point {{math|''A''}} and line {{math|''l''}} not incident with it (an '''anti-flag''') there is exactly one line {{math|''m''}} incident with {{math|''A''}} (that is, {{math|''A'' I ''m''}}), that does not meet {{math|''l''}} (known as [[Playfair's axiom]]),
and satisfying the non-degeneracy condition:
* There exists a triangle, i.e. three non-collinear points.

The lines {{math|''l''}} and {{math|''m''}} 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 {{math|''k''}}, the number of points on a line. An affine plane of order {{math|''n''}} is an {{math|((''n''<sup>2</sup>)<sub>''n'' + 1</sub>, (''n''<sup>2</sup> + ''n'')<sub>''n''</sub>)}} configuration.
[[File:Hesse configuration.svg|thumb|left|<center>Affine plane of order 3 <br> (Hesse configuration)</center>]]

====Hesse configuration====
{{main|Hesse configuration}}

The affine plane of order three is a {{math|(9<sub>4</sub>, 12<sub>3</sub>)}} configuration. When embedded in some ambient space it is called the '''[[Hesse configuration]]'''. It is not realizable in the Euclidean plane but is realizable in the [[complex projective plane]] as the nine [[inflection point]]s of an [[elliptic curve]] 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 {{math|(13<sub>4</sub>)}} 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 {{math|(8<sub>3</sub>)}} [[Mobius-Kantor configuration|Möbius–Kantor configuration]].

==Partial geometries==
[[File:GQ(2,2), the Doily.svg|thumb|Partial geometry pg(2,2,1)]]
{{main|Partial geometry}}

Given an integer {{math|''α'' ≥ 1}}, a tactical configuration satisfying:
* For every anti-flag {{math|(''B'', ''m'')}} there are {{math|''α''}} flags {{math|(''A'', ''l'')}} such that {{math|''B'' I ''l''}} and {{math|''A'' I ''m''}},
is called a ''partial geometry''. If there are {{math|''s'' + 1}} points on a line and {{math|''t'' + 1}} lines through a point, the notation for a partial geometry is {{math|pg(''s'', ''t'', ''α'')}}.

If {{math|1=''α'' = 1}} these partial geometries are [[generalized quadrangle]]s.

If {{math|1=''α'' = ''s'' +  1}} these are called [[Steiner system]]s.

==Generalized polygons==
{{main|Generalized polygon}}

For {{math|''n'' > 2}},<ref>The use of {{math|''n''}} in the name is standard and should not be confused with the number of points in a configuration.</ref> a '''generalized {{math|''n''}}-gon''' is a partial linear space whose incidence graph {{math|Γ}} has the property:
* The [[Girth (graph theory)|girth]] of {{math|Γ}} (length of the shortest [[Cycle (graph theory)|cycle]]) is twice the [[Diameter (graph theory)|diameter]] of {{math|Γ}} (the largest distance between two vertices, {{math|''n''}} 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 {{math|''n''}}-gon contains no [[Polygon|ordinary {{math|''m''}}-gon]] for {{math|2 ≤ ''m'' < ''n''}} and for every pair of objects (two points, two lines or a point and a line) there is an ordinary {{math|''n''}}-gon that contains them both.

Generalized 3-gons are projective planes. Generalized 4-gons are called [[generalized quadrangle]]s. By the Feit-Higman theorem the only finite generalized {{math|''n''}}-gons with at least three points per line and three lines per point have {{math|''n''}} = 2, 3, 4, 6 or 8.

==Near polygons==
{{main|Near polygon}}
For a non-negative integer {{math|''d''}} a '''near {{math|2''d''}}-gon''' is an incidence structure such that:
* The maximum distance (as measured in the collinearity graph) between two points is {{math|''d''}}, and
* For every point {{math|''X''}} and line {{math|''l''}} there is a unique point on {{math|''l''}} that is closest to {{math|''X''}}.

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]]. 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 [[Polar space|dual polar spaces]] are near polygons.

Many near polygons are related to [[finite simple groups]] like the [[Mathieu group]]s and the [[Janko group J2]]. Moreover, the generalized 2''d''-gons, which are related to [[Groups of Lie type]], are special cases of near 2''d''-gons.

==Möbius planes==
{{main|Möbius plane}}
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 {{math|(''P'', ''z'')}} and any point {{math|''Q''}} not incident with {{math|''z''}} there is a unique cycle {{math|''z''<sup>∗</sup>}} with {{math|''P'' I ''z''<sup>∗</sup>, ''Q'' I ''z''<sup>∗</sup>}} and {{math|1=''z'' ∩ ''z''<sup>∗</sup> =  {''P''}}}. (The cycles are said to ''touch'' at {{math|''P''}}.)
* Every cycle has at least three points and there exists at least one cycle.

The incidence structure obtained at any point {{math|''P''}} of a Möbius plane by taking as points all the points other than {{math|''P''}} and as lines only those cycles that contain {{math|''P''}} (with {{math|''P''}} removed), is an affine plane. This structure is called the ''residual'' at {{math|''P''}} in design theory.

A finite Möbius plane of ''order'' {{math|''m''}} is a tactical configuration with {{math|1=''k'' = ''m'' + 1}} points per cycle that is a [[block design|3-design]], specifically a {{math|3-(''m''<sup>2</sup> + 1, ''m'' + 1, 1)}} block design.

<!-- Does not seem to belong to this article, unsure of what to do with it

== Line–Line matrix ==
The line–line matrix indicates the number of common points for each line-pair. The line–line matrix for the Fano plane is:

[[File:Fano Line.jpg|center]]

The line–line matrix can be obtained from the incidence matrix. If ''N'' is the incidence matrix and ''N<sup>T</sup>'' is the transpose of the incidence matrix, then the line–line matrix is the matrix product, ''L'' = ''N<sup>T</sup>'' × ''N''.

== Point–Point matrix ==
The point–point matrix indicates the number of lines common to each point-pair. The point–point matrix for the Fano plane is as follows:

[[File:Fano Point.jpg|center]]

The point–point matrix can also be derived from the incidence matrix. If ''N'' is the incidence matrix and ''N<sup>T</sup>'' is the transpose of the incidence matrix, then the point–point matrix is the product ''P'' = ''N'' × ''N<sup>T</sup>''.
-->

==Incidence theorems in the Euclidean plane==

===The Sylvester-Gallai theorem===
{{main|Sylvester-Gallai theorem}}
A question raised by [[J.J. Sylvester]] in 1893 and finally settled by [[Tibor Gallai]] 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]] 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 ===
{{main|De Bruijn–Erdős theorem (incidence geometry)}}

A related result is the [[De Bruijn–Erdős theorem (incidence geometry)|de Bruijn–Erdős theorem]]. [[Nicolaas Govert de Bruijn]] and [[Paul Erdős]] proved the result in the more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is:<ref>[http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.], [http://mathworld.wolfram.com/deBruijn-ErdosTheorem.html "de Bruijn–Erdős Theorem"] from [http://mathworld.wolfram.com/ MathWorld]</ref>

::In a [[projective plane]], every non-collinear set of {{math|''n''}} points determines at least {{math|''n''}} 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 [[Mathematical induction|induction]].

===The Szemerédi–Trotter theorem ===
{{main|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 {{math|''n''}}  points and {{math|''m''}} lines in the plane, the number of flags (incident point-line pairs) is:

:<math>O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ),</math>

and this bound cannot be improved, except in terms of the implicit constants.

This result can be used to prove Beck's theorem.

===Beck's theorem===
{{main|Beck's theorem (geometry)}}

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 {{math|''C'', ''K''}} such that given any {{math|''n''}} points in the plane, at least one of the following statements is true:

# There is a line that contains at least {{sfrac|''n''|''C''}} of the points.
# There exist at least {{sfrac|''n''<sup>2</sup>|''K''}} lines, each of which contains at least two of the points.

In Beck's original argument, {{math|''C''}} is 100 and {{math|''K''}} is an unspecified constant; it is not known what the optimal values of {{math|''C''}} and {{math|''K''}} are.

== More examples ==

* [[Projective geometry|Projective geometries]]
* [[Moufang polygon]]
* [[Schläfli double six]]
* [[Reye configuration]]
* [[Cremona-Richmond configuration]]
* [[Kummer configuration]]
* [[Klein configuration]]
* [[Non-Desarguesian plane]]s

== See also ==

* [[Combinatorial design]]s
* [[Finite geometry]]
* [[Intersection theorem]]
* [[Levi graph]]

==Notes==
{{reflist}}

== References ==
*{{citation
 | last1 = Aigner | first1 = Martin
 | last2 = Ziegler | first2 = Günter M.
 | contribution = Lines in the plane and decompositions of graphs
 | doi = 10.1007/978-3-642-00856-6_10
 | location = Berlin & Heidelberg
 | pages = 63–67
 | publisher = Springer
 | title = Proofs from The Book
 | year = 2010| isbn = 978-3-642-00855-9
 }}
* {{citation|first1=Lynn Margaret|last1=Batten|author1-link=Lynn Batten|title=Combinatorics of Finite Geometries|year=1986|publisher=Cambridge University Press|place=New York|isbn=978-0-521-31857-0}}
* {{citation|first1=Lynn Margaret|last1=Batten|author1-link=Lynn Batten|first2=Albrecht|last2=Beutelspacher|author2-link=Albrecht Beutelspacher|title=The Theory of Finite Linear Spaces|year=1993|publisher=Cambridge University Press|place=New York|isbn=978-0-521-33317-7|url-access=registration|url=https://archive.org/details/theoryoffiniteli0000batt}}
* Buekenhout, Francis (1995), ''Handbook of Incidence Geometry: Buildings and Foundations'', Elsevier B.V.
* {{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=978-1-58488-506-1|edition=2nd}}
* {{cite arXiv|first1=Alberto|last1=Collino|first2=Alberto|last2=Conte|first3=Alessandro|last3=Verra|title=On the life and scientific work of Gino Fano|eprint=1311.7177|year=2013|class=math.HO}}
*{{citation| first = Bart | last = De Bruyn | title = An Introduction to Incidence Geometry | publisher = Springer International Publishing| year = 2016 | isbn = 978-3-319-43810-8 | doi = 10.1007/978-3-319-43811-5 | series = Frontiers in Mathematics}}
* {{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275  | year=1968 | isbn=978-3-540-61786-0}}
* {{cite web|last=Malkevitch|first=Joe|title=Finite Geometries?|url=http://www.ams.org/featurecolumn/archive/finitegeometries.html|accessdate=Dec 2, 2013}}
* {{cite web|last=Moorhouse|first=G. Eric|title=Incidence Geometry|url=http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf|accessdate=Oct 20, 2012}}
* {{citation
 | last = Ueberberg | first1 = Johannes
 | doi = 10.1007/978-3-642-20972-7
 | publisher = Springer
 | title = Foundations of Incidence Geometry
 | isbn = 978-3-642-26960-8
 | year = 2011
 | series = Springer Monographs in Mathematics}}.
* {{citation
 | last = Shult | first1 = Ernest E.
 | doi = 10.1007/978-3-642-15627-4
 | publisher = Springer
 | title = Points and Lines
 | isbn = 978-3-642-15626-7
 | year = 2011
 | series =Universitext}}.
* {{citation
 | last = Ball | first1 = Simeon
 | url = http://www.cambridge.org/hr/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/finite-geometry-and-combinatorial-applications
 | publisher = Cambridge University Press
 | title = Finite Geometry and Combinatorial Applications
 | isbn = 978-1107518438
 | year = 2015
 | series = London Mathematical Society Student Texts}}.

==External links==
*{{Commonscat-inline}}
*[http://www.encyclopediaofmath.org/index.php/Incidence_system ''incidence system''] at the [[Encyclopedia of Mathematics]]

[[Category:Incidence geometry| ]]