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.

摘要

我们如何在三维欧几里得空间中以最大化线对之间的最小内角的方式排列通过原点的n条线？Conway、Hardin 和 Sloane (1996) 为$n\le 55$他们推测在这个意义上是在最优的数值精度范围内，但直到现在只有情况$n\le 7$已解决。在本文中，我们解决了n  = 8 的情况。从最近关于 Tammes 问题的工作中汲取灵感，我们枚举接触图候选者以获得最佳配置，并消除那些违反各种组合和几何必要条件的候选者。Conway、Hardin 和 Sloane 的假定最优数值包装的接触图是唯一幸存的图，我们从该图中恢复了真实投影平面中八个最佳包装点的最小距离的精确表达式。