The Optimal Packing of Eight Points in the Real Projective Plane
Dustin G. Mixon, Hans Parshall · Experimental Mathematics · 2019
Dustin G. Mixon & Hans Parshall*Department of Mathematics, The Ohio State University, Columbus, OH, USACONTACT Hans Parshall [email protected] Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA.Supplemental data for this article is available online at https://doi.org/10.1080/10586458.2019.1641767.AbstractHow 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≤55 that they conjectured to be within numerical precision of optimal in this sense, but until now only the cases n≤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.