Möbius–Kantor configuration

If so, the vertices and edges of these polygons would form a projective configuration.

there is no solution in the Euclidean plane, but Seligmann Kantor (1882) 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.

That is, in Kantor's solution, the coordinates of the polygon vertices are complex numbers.

, a pair of mutually-inscribed quadrilaterals in the complex projective plane, is called the Möbius–Kantor configuration.

H. S. M. Coxeter (1950) supplies the following simple complex projective coordinates for the eight points of the Möbius–Kantor configuration: 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 3{3}3.

[2] The solution to Möbius' problem of mutually inscribed polygons for values of p greater than four is also of interest.

Both configurations may also be described algebraically in terms of the abelian group

This group has four subgroups of order three (the subsets of elements of the form

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.