The Optimal Packing of Eight Points 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 $n\le 55$ that they conjectured to be within numerical precision of optimal in this sense, but until now only the cases $n\le 7$ 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.

Keywords:

• Grassmannian manifolds
• packing
• frame theory

Acknowledgments

Both authors are grateful to the anonymous referee for many helpful comments that have greatly improved this article. We especially appreciate the suggestion of incorporating surftri in our future work.

Additional information

Funding

DGM was partially supported by AFOSR (FA9550-18-1-0107), NSF (DMS 1829955), and the Simons Institute of the Theory of Computing.