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Möbius configuration
In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two mutually inscribed tetrahedra: each vertex of one tetrahedron lies on a face 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, 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 is true more generally in a three-dimensional projective space if and only if Pappus's theorem holds for that space (Reidemeister, Schönhardt), and it is true for a three-dimensional space modeled on a division ring if and only if the ring satisfies the commutative law and is therefore a field (Al-Dhahir). By projective duality, 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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