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On Perles' configuration

Jozsef Solymosi

TL;DR

This work proves Grünbaum's conjecture for all configurations on at most eight points: any real realizable $n_3$ configuration with $n\le 8$ points also has a rational realization, making Perles' nine-point arrangement the minimal nonrationalizable example. The proof combines projective normalization with a case analysis aided by the Kelly–Moser bound on ordinary lines to show that every eight-point (and smaller) case either yields a rational realization or is unrealizable in the real plane. As applications, the paper shows that nonrational realizability can violate proposed incidence bounds for point-line configurations, and it presents Elkies' ten-point construction (no four on a line) as another obstruction to rational realizability, discussed via an elliptic-curve obstruction in the Appendix. Overall, the results illuminate the arithmetic obstructions behind planar configurations and connect dense incidence phenomena to elliptic-curve theory and modular curves.

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. Grünbaum 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.

On Perles' configuration

TL;DR

This work proves Grünbaum's conjecture for all configurations on at most eight points: any real realizable configuration with points also has a rational realization, making Perles' nine-point arrangement the minimal nonrationalizable example. The proof combines projective normalization with a case analysis aided by the Kelly–Moser bound on ordinary lines to show that every eight-point (and smaller) case either yields a rational realization or is unrealizable in the real plane. As applications, the paper shows that nonrational realizability can violate proposed incidence bounds for point-line configurations, and it presents Elkies' ten-point construction (no four on a line) as another obstruction to rational realizability, discussed via an elliptic-curve obstruction in the Appendix. Overall, the results illuminate the arithmetic obstructions behind planar configurations and connect dense incidence phenomena to elliptic-curve theory and modular curves.

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. Grünbaum 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.
Paper Structure (6 sections, 3 theorems, 3 equations, 3 figures)

This paper contains 6 sections, 3 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Grünbaum's Conjecture
  3. An application
  4. Other configurations
  5. Acknowledgements
  6. Appendix

Key Result

Theorem 1

All $n_3$ configurations are rational for $9\leq n\leq 12$.

Figures (3)

  • Figure 1: This arrangement can not be realized with points from the integer grid
  • Figure 2: Perles configuration. The labelling follows the list in ISS
  • Figure 3: Points $(6')$ and $(6")$ should be the same.

Theorems & Definitions (7)

  • Theorem 1
  • Conjecture 1: Grünbaum
  • Theorem 2: Grünbaum's Conjecture
  • proof
  • Theorem 3: Kelly-Moser
  • Conjecture 2
  • Conjecture 3