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Inmathematics,incidence geometry is the study ofincidence structures. A geometric structure such as theEuclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, andincidence. Anincidence 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.^[1]

Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects. Ingraph theory they are calledhypergraphs, and incombinatorial design theory they are calledblock designs. 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 theEuclidean 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.

Anincidence structure(P,L, I) consists of a setP whose elements are calledpoints, a disjoint setL whose elements are calledlines and anincidence relationI between them, that is, a subset ofP ×L whose elements are calledflags.^[2] If(A,l) is a flag, we say thatA isincident withl or thatl is incident withA (the terminology is symmetric), and writeA Il. Intuitively, a point and line are in this relation if and only if the point ison the line. Given a pointB and a linem which do not form a flag, that is, the point is not on the line, the pair(B,m) is called ananti-flag.

There is no natural concept of distance (ametric) in an incidence structure. However, a combinatorial metric does exist in the correspondingincidence graph (Levi graph), namely the length of the shortestpath between two vertices in thisbipartite 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 thecollinearity 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.

Incidence structures that are most studied are those that satisfy some additional properties (axioms), such asprojective planes,affine planes,generalized polygons,partial geometries andnear polygons. Very general incidence structures can be obtained by imposing "mild" conditions, such as:

Apartial linear space is an incidence structure for which the following axioms are true:^[3]

• 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 byk.

RPr: Each point is incident with the same number of lines. If finite this number is often denoted byr.

The second axiom of a partial linear space implies thatk > 1. Neither regularity condition implies the other, so it has to be assumed thatr > 1.

A finite partial linear space satisfying both regularity conditions withk,r > 1 is called atactical configuration.^[4] Some authors refer to these simply asconfigurations,^[5] orprojective configurations.^[6] If a tactical configuration hasn points andm lines, then, by double counting the flags, the relationshipnr =mk is established. A common notation refers to(n_r,m_k)-configurations. In the special case wheren =m (and hence,r =k) the notation(n_k,n_k) is often simply written as(n_k).

Alinear space is a partial linear space such that:^[7]

• 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.^[8]

This is used to rule out some very small examples (mainly when the setsP orL 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 beingtrivial and those that do asnon-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 aSylvester–Gallai design.

Some of the basic concepts and terminology arises from geometric examples, particularlyprojective planes andaffine planes.

Aprojective 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 arecollinear.

There is abijection betweenP andL in a projective plane. IfP is a finite set, the projective plane is referred to as afinite projective plane. Theorder of a finite projective plane isn =k – 1, that is, one less than the number of points on a line. All known projective planes have orders that areprime powers. A projective plane of ordern is an((n² +n + 1)_{n + 1}) configuration.

The smallest projective plane has order two and is known as theFano plane.

This famous incidence geometry was developed by the Italian mathematicianGino Fano. In his work^[9] on proving the independence of the set of axioms forprojectiven-space that he developed,^[10] 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.^[11] The planes in this space consisted of seven points and seven lines and are now known asFano planes.

The Fano plane cannot be represented in theEuclidean plane using only points and straight line segments (i.e., it is not realizable). This is a consequence of theSylvester–Gallai theorem, according to which every realizable incidence geometry must include anordinary line, a line containing only two points. The Fano plane has no such line (that is, it is aSylvester–Gallai configuration), so it is not realizable.^[12]

Acomplete 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 theFano axiom, often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear.

Anaffine plane is a linear space satisfying:

• For any pointA and linel not incident with it (ananti-flag) there is exactly one linem incident withA (that is,A Im), that does not meetl (known asPlayfair's axiom),

and satisfying the non-degeneracy condition:

• There exists a triangle, i.e. three non-collinear points.

The linesl andm in the statement of Playfair's axiom are said to beparallel. Every affine plane can be uniquely extended to a projective plane. Theorder of a finite affine plane isk, the number of points on a line. An affine plane of ordern is an((n²)_{n + 1}, (n² +n)_n) configuration.

The affine plane of order three is a(9₄, 12₃) configuration. When embedded in some ambient space it is called theHesse configuration. It is not realizable in the Euclidean plane but is realizable in thecomplex projective plane as the nineinflection points of anelliptic 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 calledparallel 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(13₄) 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(8₃)Möbius–Kantor configuration.

Given an integerα ≥ 1, a tactical configuration satisfying:

• For every anti-flag(B,m) there areα flags(A,l) such thatB Il andA Im,

is called apartial geometry. If there ares + 1 points on a line andt + 1 lines through a point, the notation for a partial geometry ispg(s,t,α).

Ifα = 1 these partial geometries aregeneralized quadrangles.

Ifα =s + 1 these are calledSteiner systems.

Forn > 2,^[13] ageneralizedn-gon is a partial linear space whose incidence graphΓ has the property:

• Thegirth ofΓ (length of the shortestcycle) is twice thediameter ofΓ (the largest distance between two vertices,n in this case).

Ageneralized 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 generalizedn-gon contains noordinarym-gon for2 ≤m <n and for every pair of objects (two points, two lines or a point and a line) there is an ordinaryn-gon that contains them both.

Generalized 3-gons are projective planes. Generalized 4-gons are calledgeneralized quadrangles. By the Feit-Higman theorem the only finite generalizedn-gons with at least three points per line and three lines per point haven = 2, 3, 4, 6 or 8.

For a non-negative integerd anear2d-gon is an incidence structure such that:

• The maximum distance (as measured in the collinearity graph) between two points isd, and
• For every pointX and linel there is a unique point onl that is closest toX.

A near 0-gon is a point, while a near 2-gon is a line. The collinearity graph of a near 2-gon is acomplete 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, alldual polar spaces are near polygons.

Many near polygons are related tofinite simple groups like theMathieu groups and theJanko group J2. Moreover, the generalized 2d-gons, which are related toGroups of Lie type, are special cases of near 2d-gons.

An abstract Möbius 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 ascycles orblocks.

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(P,z) and any pointQ not incident withz there is a unique cyclez^∗ withP Iz^∗,Q Iz^∗ andz ∩z^∗ = {P}. (The cycles are said totouch atP.)
• Every cycle has at least three points and there exists at least one cycle.

The incidence structure obtained at any pointP of a Möbius plane by taking as points all the points other thanP and as lines only those cycles that containP (withP removed), is an affine plane. This structure is called theresidual atP in design theory.

A finite Möbius plane oforderm is a tactical configuration withk =m + 1 points per cycle that is a3-design, specifically a3-(m² + 1,m + 1, 1) block design.

A question raised byJ.J. Sylvester in 1893 and finally settled byTibor 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 eithercollinear or there exists a line incident with exactly two of the points.

A line containing exactly two of the points is called anordinary line in this context. Sylvester was probably led to the question while pondering about the embeddability of the Hesse configuration.

A related result is thede Bruijn–Erdős theorem.Nicolaas Govert de Bruijn andPaul Erdős proved the result in the more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is:^[14]

In aprojective plane, every non-collinear set ofn points determines at leastn 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 usinginduction.

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): givenn points andm lines in the plane, the number of flags (incident point-line pairs) is:

${\displaystyle O\left(n^{2/3}{m}^{2/3}+n+m\right),}$

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.^[15]

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 constantsC,K such that given anyn points in the plane, at least one of the following statements is true:

1. There is a line that contains at least⁠n/C⁠ of the points.
2. There exist at least⁠n²/K⁠ lines, each of which contains at least two of the points.

In Beck's original argument,C is 100 andK is an unspecified constant; it is not known what the optimal values ofC andK are.

• Projective geometries
• Moufang polygon
• Schläfli double six
• Reye configuration
• Cremona–Richmond configuration
• Kummer configuration
• Klein configuration
• Non-Desarguesian planes

• Combinatorial designs
• Finite geometry
• Intersection theorem
• Levi graph

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• Malkevitch, Joe."Finite Geometries?". RetrievedDec 2, 2013.
• Moorhouse, G. Eric."Incidence Geometry"(PDF). Archived fromthe original(PDF) on October 29, 2013. RetrievedOct 20, 2012.
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