Configurations (geometry) | Polyhedral combinatorics
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, omitting one of the symmetry lines. 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. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations. (Wikipedia).
