In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the Möbius–Kantor configuration is a
configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice board ...
consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of
incidences in the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, but it is possible in the
complex projective plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...
.
Coordinates
asked whether there exists a pair of
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s with ''p'' sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a
projective configuration. For
there is no solution in the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the
complex projective plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...
. That is, in Kantor's solution, the coordinates of the polygon vertices are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. Kantor's solution for
, a pair of mutually-inscribed quadrilaterals in the complex projective plane, is called the Möbius–Kantor configuration.
supplies the following simple
complex projective coordinates for the eight points of the Möbius–Kantor configuration:
:(1,0,0), (0,0,1), (ω, −1, 1), (−1, 0, 1),
:(−1,ω
2,1), (1,ω,0), (0,1,0), (0,−1,1),
where ω denotes a complex
cube root of 1.
The eight points and eight lines of the Möbius–Kantor configuration, with these coordinates, form the eight vertices and eight 3-edges of the
complex polygon
The term ''complex polygon'' can mean two different things:
* In geometry, a polygon in the unitary plane, which has two complex dimensions.
* In computer graphics, a polygon whose boundary is not simple.
Geometry
In geometry, a complex pol ...
33. Coxeter named it a
Möbius–Kantor polygon.
Abstract incidence pattern
More abstractly, the Möbius–Kantor configuration can be described as a system of eight points and eight triples of points such that each point belongs to exactly three of the triples. With the additional conditions (natural to points and lines) that no pair of points belong to more than one triple and that no two triples have more than one point in their intersection, any two systems of this type are equivalent under some
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
of the points. That is, the Möbius–Kantor configuration is the unique
projective configuration of type (8
38
3).
The
Möbius–Kantor graph
In the mathematics, mathematical field of graph theory, the Möbius–Kantor graph is a symmetric graph, symmetric bipartite graph, bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It ...
derives its name from being the
Levi graph of the Möbius–Kantor configuration. It has one vertex per point and one vertex per triple, with an edge connecting two vertices if they correspond to a point and to a triple that contains that point.
The points and lines of the Möbius–Kantor configuration can be described as a
matroid
In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...
, whose elements are the points of the configuration and whose nontrivial flats are the lines of the configuration. In this matroid, a set ''S'' of points is independent if and only if either
or ''S'' consists of three non-collinear points. As a matroid, it has been called the MacLane matroid, after the work of proving that it cannot be
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is ori ...
; it is one of several known
minor-minimal non-orientable matroids.
Related configurations
The solution to Möbius' problem of mutually inscribed polygons for values of ''p'' greater than four is also of interest. In particular, one possible solution for
is the
Desargues configuration
In geometry, the Desargues configuration is a Configuration (geometry), configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues.
The Desargues configuration can be const ...
, a set of ten points and ten lines, three points per line and three lines per point, that does admit a Euclidean realization. The
Möbius configuration
In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was ...
is a three-dimensional analogue of the Möbius–Kantor configuration consisting of two mutually inscribed tetrahedra.
The Möbius–Kantor configuration can be augmented by adding four lines through the four pairs of points not already connected by lines, and by adding a ninth point on the four new lines. The resulting configuration, the
Hesse configuration
In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be denoted as (94 123) or configuration matrix \left begin9 & 4 \\ 3 & 12 \\ \end\right /math>. ...
, shares with the Möbius–Kantor configuration the property of being realizable with complex coordinates but not with real coordinates.
[.] Deleting any one point from the Hesse configuration produces a copy of the Möbius–Kantor configuration.
Both configurations may also be described algebraically in terms of the
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
with nine elements.
This group has four subgroups of order three (the subsets of elements of the form
,
,
, and
respectively), each of which can be used to partition the nine group elements into three
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of three elements per coset. These nine elements and twelve cosets form the Hesse configuration. Removing the zero element and the four cosets containing zero gives rise to the Möbius–Kantor configuration.
Notes
References
*.
*. Reprinted in ''The Visual Mind'', MIT Press, 1993, pp. 19–26, .
*.
*.
*.
*. In ''Gesammelte Werke'' (1886), vol. 1, pp. 439–446.
*.
External links
*
{{DEFAULTSORT:Mobius-Kantor configuration
Configurations (geometry)