论文标题
在八维
Enriques surfaces and an Apollonian packing in eight dimensions
论文作者
论文摘要
如果球体切向或根本没有相交，我们将$ n $尺寸中的超球包装称为apollonian球体。他们填补了$ n $维的空间；包装中的每个球体都是$ N+2 $相互切线球（以及本文中描述的更多属性）的群集的成员。在本文中，我们描述了一种在八个维度上的阿波罗尼亚人包装，这自然源于对通用淋巴结的研究。 $ E_7 $，$ E_8 $和Reye Lattices扮演角色。我们使用包装在九个维度上生成一个apollonian包装，而在七个维度上的横截面却是微弱的apollonian。麦克斯韦描述了这三个包装，但似乎没有意识到它们是阿波罗尼亚人。七个和八个维度的包装与早期论文中的包装不同。顺便说一句，我们提供了足够的条件，可以使Coxeter图生成相互切线球，并使用它来识别三个维度的Apollonian球体堆积不是肥大的球体堆积。
We call a packing of hyperspheres in $n$ dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the $n$-dimensional space; and every sphere in the packing is a member of a cluster of $n+2$ mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The $E_7$, $E_8$ and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres, and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.
