Perles configuration

In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates.

It can be constructed from the diagonals and symmetry lines of a regular pentagon, and their crossing points.

In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example.

One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals.

[1] A realization of the Perles configuration is defined to consist of any nine points and nine lines with the same intersection pattern.

Every realization of this configuration in the Euclidean plane or, more generally, in the real projective plane is equivalent, under a projective transformation, to a realization constructed in this way from a regular pentagon.

Mac Lane (1936) describes an 11-point example, obtained by applying Von Staudt's algebra of throws to construct a configuration corresponding to the square root of two.