On Perles' configuration

TL;DR

This paper proves Gr"unbaum's conjecture that any realizable geometric arrangement of eight or fewer points in the plane can be realized with rational coordinates, confirming the minimality of Perles' nine-point configuration.

Contribution

The paper provides a proof that Perles' nine-point configuration is the smallest non-rational realizable arrangement in the plane.

Findings

Perles' configuration cannot be realized with rational coordinates.

Any realizable arrangement of eight or fewer points can be realized with rational coordinates.

The conjecture by Gr"unbaum is confirmed.

Abstract

In the 60s, Micha Perles constructed a point-line arrangement in the plane on nine points, which can not be realized only by points with rational coordinates. Gr\"unbaum conjectured that Perles' construction is the smallest: any geometric arrangement on eight or fewer points if it is realizable with real coordinates in the plane, it is also realizable with rational coordinates. In this paper, we prove the conjecture.