The optimal packing of eight points in the real projective plane

TL;DR

This paper determines the optimal arrangement of eight points in the real projective plane by enumerating and eliminating candidate configurations, ultimately identifying the unique optimal packing and its exact minimal distance.

Contribution

It solves the previously open case of eight points in the real projective plane, providing an exact configuration and minimal distance through combinatorial and geometric analysis.

Findings

Identified the unique contact graph for optimal packing of eight points.

Derived an exact expression for the minimal distance in the optimal configuration.

Extended understanding of point packings in the real projective plane.

Abstract

How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for $n \leq 55$ that they conjectured to be within numerical precision of optimal in this sense, but until now only the cases $n \leq 7$ have been solved. In this paper, we resolve the case $n = 8$. Drawing inspiration from recent work on the Tammes problem, we enumerate contact graph candidates for an optimal configuration and eliminate those that violate various combinatorial and geometric necessary conditions. The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and we recover from this graph an exact expression for the minimum distance of eight optimally packed points in the real projective plane.