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 configuration or Möbius tetrads is a certain

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 ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

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 ...

, consisting of two

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

that are mutually

inscribed An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same th ...

: each vertex of one tetrahedron lies on a

face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...

plane of the other tetrahedron and vice versa. Thus, for the resulting system of eight points and eight planes, each point lies on four planes (the three planes defining it as a vertex of a tetrahedron and the fourth plane from the other tetrahedron that it lies on), and each plane contains four points (the three tetrahedron vertices of its face, and the vertex from the other tetrahedron that lies on it).

Möbius's theorem

The configuration is named after

August Ferdinand Möbius August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer. Life and education Möbius was born in Schulpforta, Electorate of Saxony, and was descended on his mothe ...

, who in 1828 proved that, if two tetrahedra have the property that seven of their vertices lie on corresponding face planes of the other tetrahedron, then the eighth vertex also lies on the plane of its corresponding face, forming a configuration of this type. This

incidence theorem In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects ...

is true more generally in a three-dimensional projective space if and only if Pappus's theorem holds for that space (

Reidemeister Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...

, Schönhardt), and it is true for a three-dimensional space modeled on a

division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...

if and only if the ring satisfies the

commutative law In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...

and is therefore a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

(Al-Dhahir). By

projective duality In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one th ...

, Möbius' result is equivalent to the statement that, if seven of the eight face planes of two tetrahedra contain the corresponding vertices of the other tetrahedron, then the eighth face plane also contains the same vertex.

Construction

describes a simple construction for the configuration. Beginning with an arbitrary point in Euclidean space, let be four planes through , no three of which share a common intersection line, and place the six points on the six lines formed by pairwise intersection of these planes in such a way that no four of these points are coplanar. For each of the planes , four of the seven points lie on that plane and three are disjointed from it; form planes through the triples of points disjoint from respectively. Then, by the dual form of Möbius' theorem, these four new planes meet in a single point . The eight points and the eight planes form an instance of Möbius' configuration.

Related constructions

state (without references) that there are five configurations having eight points and eight planes with four points on every plane and four planes through every point that are realisable in three-dimensional Euclidean space: such configurations have the shorthand notation . They must have obtained their information from the article by . This actually states, depending upon results by , , and , that there are five configurations with the property that at most two planes have two points in common, and dually at most two points are common to two planes. (This condition means that every three points may be non-collinear and dually three planes may not have a line in common.) However, there are ten other configurations that do not have this condition, and all fifteen configurations are realizable in real three-dimensional space. The configurations of interest are those with two tetrahedra, each inscribing and circumscribing the other, and these are precisely those that satisfy the above property. Thus, there are five configurations with tetrahedra, and they correspond to the five conjugacy classes of the symmetric group . One obtains a permutation from the four points of one tetrahedron to itself as follows: each point of is on a plane containing three points of the second tetrahedron . This leaves the other point of , which is on three points of a plane of , leaving another point of , and so the permutation maps . The five

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...

es have representatives and, of these, the Möbius configuration corresponds to the conjugacy class . It could be denoted . It is stated by Steinitz that if two of the complementary tetrahedra of are , and then the eight planes are given by with odd, while the even sums and their complements correspond to all pairs of complementary tetrahedra that in- and circumscribe in the model of . It is also stated that by Steinitz that the only that is a geometrical theorem is the Möbius configuration. However that is disputed: shows using a computer search and proofs that there are precisely two that are actually "theorems": the Möbius configuration and one other. The latter (which corresponds to the conjugacy class above) is also a theorem for all three-dimensional projective spaces over a

, but not over a general

. There are other close similarities between the two configurations, including the fact that both are self-dual under

Matroid duality 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 ...

. In abstract terms, the latter configuration has "points" and "planes" , where these integers are modulo eight. This configuration, like Möbius, can also be represented as two tetrahedra, mutually inscribed and circumscribed: in the integer representation the tetrahedra can be and . However, these two configurations are non-isomorphic, since Möbius has four pairs of disjoint planes, while the latter one has no disjoint planes. For a similar reason (and because pairs of planes are degenerate quadratic surfaces), the Möbius configuration is on more quadratic surfaces of three-dimensional space than the latter configuration. 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 ...

of the Möbius configuration has 16 vertices, one for each point or plane of the configuration, with an edge for every incident point-plane pair. It is isomorphic to the 16-vertex

hypercube graph In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cubical graph, cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has ...

. A closely related configuration, the

Möbius–Kantor configuration In geometry, the Möbius–Kantor configuration is a configuration 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 i ...

formed by two mutually inscribed quadrilaterals, has 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 ...

, a subgraph of , as its Levi graph.

References

*. *. *. *. *. *. *. In ''Gesammelte Werke'' (1886), vol. 1, pp. 439–446. *. *. *. *. {{DEFAULTSORT:Mobius Configuration Configurations (geometry)