![Desargues theorem alt](https://upload.wikimedia.org/wikipedia/commons/c/c8/Desargues_theorem_alt.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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
Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are ...
.
The Desargues configuration can be constructed in two dimensions from the points and lines occurring in
Desargues's 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 ...
, in three dimensions from five planes in
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
, or in four dimensions from the
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
, the four-dimensional regular
simplex. It has a large group of symmetries, taking any point to any other point and any line to any other line. It is also self-dual, meaning that if the points are replaced by lines and vice versa using
projective duality
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of du ...
, the same configuration results.
Graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
associated with the Desargues configuration include the
Desargues graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level ...
(its graph of point-line incidences) and the
Petersen graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is n ...
(its graph of non-incident lines). The Desargues configuration is one of ten different configurations with ten points and lines, three points per line, and three lines per point, nine of which can be realized in the
Euclidean plane.
Constructions
Two dimensions
Two triangles
and
are said to be in perspective centrally if the lines
,
, and
meet in a common point, called the ''center of perspectivity''. They are in perspective axially if the intersection points of the corresponding triangle sides,
,
, and
all lie on a common line, the ''axis of perspectivity''.
Desargues's 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 ...
in geometry states that these two conditions are equivalent: if two triangles are in perspective centrally then they must also be in perspective axially, and vice versa. When this happens, the ten points and ten lines of the two perspectivities (the six triangle vertices, three crossing points, and center of perspectivity, and the six triangle sides, three lines through corresponding pairs of vertices, and axis of perspectivity) together form an instance of the Desargues configuration.
Three dimensions
Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration. This construction is closely related to the property that every
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 ...
that can be embedded into a 3-dimensional projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron.
Four dimensions
![5-cell](https://upload.wikimedia.org/wikipedia/commons/d/d8/5-cell.gif)
The
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
or pentatope (a regular
simplex in four dimensions) has five
vertices, ten
edges, ten triangular
ridges
A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...
(2-dimensional faces), and five tetrahedral
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
; the edges and ridges touch each other in the same pattern as the Desargues configuration. Extend each of the edges of the 5-cell to the line that contains it (its
affine hull
In mathematics, the affine hull or affine span of a set ''S'' in Euclidean space R''n'' is the smallest affine set containing ''S'', or equivalently, the intersection of all affine sets containing ''S''. Here, an ''affine set'' may be defined ...
), similarly extend each triangle of the 5-cell to the 2-dimensional plane that contains it, and intersect these lines and planes by a three-dimensional
hyperplane that neither contains nor is parallel to any of them. Each line intersects the hyperplane in a point, and each plane intersects the hyperplane in a line; these ten points and lines form an instance of the Desargues configuration.
Symmetries
Although Desargues' theorem chooses different roles for its ten lines and points, the Desargues configuration itself is more
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
: ''any'' of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity. The Desargues configuration has a
symmetry group of order 120; that is, there are 120 different ways of permuting the points and lines of the configuration in a way that preserves its point-line incidences. The three-dimensional construction of the Desargues configuration makes these symmetries more readily apparent: if the configuration is generated from five planes in general position in three dimensions, then each of the 120 different
permutations of these five planes corresponds to a symmetry of the configuration.
The Desargues configuration is self-dual, meaning that it is possible to find a correspondence from points of one Desargues configuration to lines of a second configuration, and from lines of the first configuration to points of a second configuration, in such a way that all of the configuration's incidences are preserved.
Graphs
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 Desargues configuration, a graph having one vertex for each point or line in the configuration, is known as the
Desargues graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level ...
. Because of the symmetries and self-duality of the Desargues configuration, the Desargues graph is a
symmetric graph
In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism
:f : V(G) \rightarrow V(G)
such that
:f(u_1) = u_2 and f(v_1) = v_2.
In oth ...
.
![Petersen2 tiny](https://upload.wikimedia.org/wikipedia/commons/9/97/Petersen2_tiny.svg)
draws a different graph for this configuration, with ten vertices representing its ten lines, and with two vertices connected by an edge whenever the corresponding two lines do not meet at one of the points of the configuration. Alternatively, the vertices of this graph may be interpreted as representing the points of the Desargues configuration, in which case the edges connect pairs of points for which the line connecting them is not part of the configuration. This publication marks the first known appearance of the
Petersen graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is n ...
in the mathematical literature, 12 years before
Julius Petersen
Julius Peter Christian Petersen (16 June 1839, Sorø, West Zealand – 5 August 1910, Copenhagen) was a Danish mathematician. His contributions to the field of mathematics led to the birth of graph theory.
Biography
Petersen's interests i ...
's use of the same graph as a counterexample to an
edge coloring
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blu ...
problem.
Related configurations
![Non-Desargues configuration](https://upload.wikimedia.org/wikipedia/commons/f/f8/Non-Desargues_configuration.svg)
As a projective configuration, the Desargues configuration has the notation (10
310
3), meaning that each of its ten points is incident to three lines and each of its ten lines is incident to three points. Its ten points can be viewed in a unique way as a pair of mutually inscribed
pentagons, or as a self-inscribed
decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
. The
Desargues graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level ...
, a 20-vertex
bipartite symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
cubic graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.
A bicubic graph is a cubic bi ...
, is so called because it can be interpreted as 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 Desargues configuration, with a vertex for each point and line of the configuration and an edge for every incident point-line pair.
There also exist eight other (10
310
3) configurations (that is, sets of points and lines in the Euclidean plane with three lines per point and three points per line) that are not
incidence-isomorphic to the Desargues configuration, one of which is shown at right. A tenth configuration exists as an abstract
finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
but cannot be realized using Euclidean points and lines. In all of these configurations, each point has three other points that are not collinear with it. But in the Desargues configuration, these three points are always collinear with each other (if the chosen point is the center of perspectivity, then the three points form the axis of perspectivity) while in the other configuration shown in the illustration these three points form a triangle of three lines. As with the Desargues configuration, the other depicted configuration can be viewed as a pair of mutually inscribed pentagons.
[This configuration is the ''cyclic'' 103 ''configuration'', part of a family of configurations studied by .]
Notes
References
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External links
*
{{Incidence structures
Configurations (geometry)