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 c ...
, the Reye configuration, introduced by , 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 ...
of 12
points and 16
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye configuration is written as .
Realization
The Reye configuration can be realized in three-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
by taking the lines to be the 12 edges and four long diagonals of a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, and the points as the eight vertices of the cube, its center, and the three points where groups of four parallel cube edges meet the plane at infinity. Two regular
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
may be inscribed within a cube, forming a
stella octangula
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted ...
; these two tetrahedra are perspective figures to each other in four different ways, and the other four points of the configuration are their centers of perspectivity. These two tetrahedra together with the tetrahedron of the remaining 4 points form a
desmic system
Two desmic tetrahedra. The third tetrahedron of this system is not shown, but has one vertex at the center and the other three on the plane at infinity.
In projective geometry, a desmic system () is a set of three tetrahedra in 3-dimensional p ...
of three tetrahedra.
Any two disjoint spheres in three dimensional space, with different radii, have two
bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at .
Bitangents of algebraic curves
In general, an algebraic cur ...
double cones, the apexes of which are called the centers of similitude. If three spheres are given, with their centers non-collinear, then their six centers of similitude form the six points of a
complete quadrilateral
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
, the four lines of which are called the axes of similitude. And if four spheres are given, with their centers non-coplanar, then they determine 12 centers of similitude and 16 axes of similitude, which together form an instance of the Reye configuration .
The Reye configuration can also be realized by points and lines in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, by drawing the three-dimensional configuration in
three-point perspective
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
. An 8
312
2 configuration of eight points in the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
and 12 lines connecting them, with the connection pattern of a cube, can be extended to form the Reye configuration if and only if the eight points are a
perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
of a
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
The 24 permutations of the points
form the vertices of a
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
centered at the origin of four-dimensional Euclidean space.
These 24 points also form the 24 roots in the
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
.
They can be grouped into pairs of points opposite each other on a line through the origin. The lines and planes through the origin of four-dimensional Euclidean space have the geometry of the points and lines of three-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, and in this three-dimensional projective space the lines through opposite pairs of these 24 points and the central planes through these points become the points and lines of the Reye configuration . The permutations of
form the
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
of the 12 points in this configuration.
Application
pointed out that the Reye configuration underlies some of the proofs of the
Bell–Kochen–Specker theorem about the non-existence of hidden variables in quantum mechanics.
Related configurations
The
Pappus configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point.
History and construction
This configuration is named after Pappus of ...
may be formed from two triangles that are perspective figures to each other in three different ways, analogous to the interpretation of the Reye configuration involving desmic tetrahedra.
If the Reye configuration is formed from a cube in three-dimensional space, then there are 12 planes containing four lines each: the six face planes of the cube, and the six planes through pairs of opposite edges of the cube. Intersecting these 12 planes and 16 lines with another plane in general position produces a 16
312
4 configuration, the dual of the Reye configuration. The original Reye configuration and its dual together form a 28
428
4 configuration .
There are 574 distinct configurations of type 12
416
3 .
References
*
*
*.
*.
*. See also pp. 154–157.
*. See in particular section 2.1, "The Reye configuration and triality", pp. 460–461.
*.
*.
{{Incidence structures
Configurations (geometry)
Polyhedral combinatorics