Incidence geometry (structure)
Encyclopedia
Incidence geometry is an area of mathematics
that studies relations of incidence
between various geometrical objects such as points, lines, curves, and planes. A specific collection of such objects is called a mathematical structure
. One type of mathematical structure of particular importance contains only points and lines and is called an incidence geometry.
The result of this is that every incidence geometry contains at least three points and three lines. Thus, the simplest incidence geometry that can exist would look something like this:
and is known as the Fano plane
:
which serves as a visual representation of all incidence relations in the geometry. The rows of the matrix represent points, while the columns represent lines. The incidence matrix for the Fano plane looks like this:
The incidence matrix shows the sets of points and lines and which points and lines are incident. In most cases, this is sufficient information to determine the entire geometry, which is one reason why the study of incidence geometry is important.
The line-line matrix can be derived from the incidence matrix. If N is the incidence matrix and NT is the transpose of the incidence matrix, then the line-line matrix L = NT × N.
The point-point matrix can also be derived from the incidence matrix. If N is the incidence matrix and NT is the transpose of the incidence matrix, then the point-point matrix P = N × NT.
is an important theorem in the field of incidence geometry. It was proposed by two mathematicians, Nicolaas Govert de Bruijn
and Paul Erdös
. The statement of the theorem is as follows:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that studies relations of incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
between various geometrical objects such as points, lines, curves, and planes. A specific collection of such objects is called a mathematical structure
Mathematical structure
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....
. One type of mathematical structure of particular importance contains only points and lines and is called an incidence geometry.
Definition
An incidence geometry is an incidence structure for which the following axioms are true:- Every pair of distinct points determines a unique line.
- Every line contains at least two distinct points.
- For every line, there is at least one point that does not lie on the line.
The result of this is that every incidence geometry contains at least three points and three lines. Thus, the simplest incidence geometry that can exist would look something like this:
The Fano Plane
One famous incidence geometry was developed by the Italian mathematician FanoFano
Fano is a town and comune of the province of Pesaro and Urbino in the Marche region of Italy. It is a beach resort 12 km southeast of Pesaro, located where the Via Flaminia reaches the Adriatic Sea...
and is known as the Fano plane
Fano plane
In finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.-Homogeneous coordinates:...
:
Incidence Matrix
An incidence geometry can be modeled by an incidence matrixIncidence matrix
In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related ...
which serves as a visual representation of all incidence relations in the geometry. The rows of the matrix represent points, while the columns represent lines. The incidence matrix for the Fano plane looks like this:
The incidence matrix shows the sets of points and lines and which points and lines are incident. In most cases, this is sufficient information to determine the entire geometry, which is one reason why the study of incidence geometry is important.
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 as follows:The line-line matrix can be derived from the incidence matrix. If N is the incidence matrix and NT is the transpose of the incidence matrix, then the line-line matrix L = NT × 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:The point-point matrix can also be derived from the incidence matrix. If N is the incidence matrix and NT is the transpose of the incidence matrix, then the point-point matrix P = N × NT.
The de Bruijn-Erdös Theorem
The de Bruijn-Erdös theoremDe Bruijn–Erdős theorem (incidence geometry)
In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős, states a lower bound on the number of lines determined by n points in a projective plane...
is an important theorem in the field of incidence geometry. It was proposed by two mathematicians, Nicolaas Govert de Bruijn
Nicolaas Govert de Bruijn
Nicolaas Govert de Bruijn is a Dutch mathematician, affiliated as professor emeritus with the Eindhoven University of Technology. He received his Ph.D. in 1943 from Vrije Universiteit Amsterdam....
and Paul Erdös
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
. The statement of the theorem is as follows:
- Every non-collinear set of n points determines at least n distinct lines.
More Examples
- Affine planesAffine plane (incidence geometry)In incidence geometry, an affine plane is a system of points and lines that satisfy the following axioms :* Any two distinct points lie on a unique line.* Given a point and line there is a unique line which contains the point and is parallel to the line...
- Projective geometriesProjective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
- Möbius geometries
- Polar spacePolar spaceIn mathematics, in the field of combinatorics, a polar space of rank n , or projective index n−1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms :* Every subspace, together with its own subspaces, is isomorphic...
s - Finite geometriesFinite geometryA finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
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s - Projective configuration
- Levi graphLevi graphIn combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for...