球面码的刚性
Rigidity of spherical codes
摘要：
在球体表面上的球形帽的填充物（即球形码）如果在填充物的空间内被隔离，则称为刚性或堵塞。换句话说，除了应用全局等距外，填充不能变形。本文系统地研究了球码的刚性，特别是接吻构形。一个令人惊讶的是，考克斯特-托德晶格的接吻构型并没有被卡住，尽管被局部卡住了（如果它的邻居是固定的，每个单独的帽都被固定在适当的位置）;在这方面，考克斯特-托德晶格类似于三维的面心立方晶格。相比之下，我们发现许多其他填充都有拥挤的接吻构型，包括巴恩斯-沃尔晶格和所有已知的四维到十二维的最佳接吻构型。在更高的维度中，对于大的吻接构型，干扰似乎变得不那么常见，特别是对于25到31维中已知的最好的吻接构型，它失败了。受这种现象的启发，我们发现了新的接吻配置在这些方面，这改善了记录在1982年由层压晶格。
英文摘要：
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.
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