Möbius configuration

In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two tetrahedra that are mutually inscribed: each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa.

The configuration is named after August Ferdinand Möbius, 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.

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.

It is stated by Steinitz that if two of the complementary tetrahedra of Ke are A0, B0, C0, D0, and A1, B1, C1, D1 then the eight planes are given by Ai, Bj, Ck, Dl with i + j + k + l odd, while the even sums and their complements correspond to all pairs of complementary tetrahedra that in- and circumscribe in the model of Ke.

However that is disputed: Glynn (2010) shows using a computer search and proofs that there are precisely two S4 that are actually "theorems": the Möbius configuration and one other.

The latter (which corresponds to the conjugacy class (12)(34) above) is also a theorem for all three-dimensional projective spaces over a field, but not over a general division ring.