The Symmetry of Densest Packings of Space

这个建议是在希尔伯特第十八问题的精神，研究欧几里得空间和双曲空间的最优密集填充的对称性，由球体或多面体组成。很少有这样的包装问题已经被解决了，但我们的目标是更直接地研究这样的问题作为一个类，指定它们，使它们是适定的，具有合理的存在和唯一性条件，特别是研究解决方案的对称性。这项工作的部分动机是发现非周期平铺，如彭罗斯平铺的风筝和飞镖多边形的平面，平铺可以理解为这样一个包装问题的解决方案。平铺空间的对称性长期以来与遍历论的数学联系在一起，使用平铺空间的平移群或全同余群作为动力学。然后，将平铺的对称性与相关动力系统的共轭类联系起来。本文提出了关于包装问题的对称性的几个问题，一些是针对一般定性行为的，一些是针对重要的特殊情况的。例如，提出了非双曲空间中最密集的固定半径球体的对称在不同的传统下是不同的。（我们已经知道，对于大多数半径来说，密度最大的填料都是非周期性的——它们不可能具有晶体对称。）更普遍的是，它提出了一个“一般的”填充问题，核里得空间或双曲空间，只有最优解不是晶体学的。任何试图把尽可能多的硬币挤在桌面上的人都发现，最有效的安排也是非常对称的，每个硬币周围有6个硬币。空间中球体的有效填充的类似问题也导致了高对称性。但是，一般来说，人们几乎不知道为什么效率会导致对称，以及什么样的对称是可能的。二十年前，人们发现了一种新的金属合金，这是一个与之密切相关的最优化问题的物理解决方案。人们发现，这种合金具有对称性，但这种对称性的性质不太明显，或者换句话说，对称性的数学计算还不太完善。这一建议涉及球体和多面体空间中有效排列的对称性研究，特别是发展一种数学形式，可以有效地分析这种对称性。还规定了有关球体有效填料性质的特殊问题。

This proposal is in the spirit of Hilbert's Eighteenth problem, tostudy the symmetries of the optimally dense packings, by spheres orpolyhedra, of Euclidean and hyperbolic spaces of general dimension.Very few particular such packing problems have ever been solved, butour goal is more directed to study such problems as a class, tospecify them so that they are well-posed, with reasonable conditionsfor existence and uniqueness, and in particular to then study thesymmetries of the solutions. This work is motivated in part by thediscovery of aperiodic tilings, such as the Penrose tilings of theplane by the kite and dart polygons, tilings which can be understoodas solutions of such a packing problem. The symmetries of aperiodictilings have long been connected with the mathematics of ergodictheory, using either the translation group or the full congruencegroup of the space being tiled as the dynamics. The symmetry of thetilings has then been related to the conjugacy class of the associateddynamical system. Several questions are proposed here about thesymmetry of packing problems, some directed at general qualitativebehavior and some directed at important special cases. For instance,it is proposed to show that the symmetry of the densest packings of ahyperbolic space by spheres of fixed radius is different for differentradii. (It is already known that for most radii the densest packingsare aperiodic - they cannot have crystallographic symmetry.) And moregenerally it is proposed to show that a "generic" packing problem, inEuclidean or hyperbolic space, only has optimal solutions which arenot crystallographic.Anyone who has tried to squeeze as many pennies as possible onto atabletop has seen that the most efficient arangement is also verysymmetrical, with six pennies surrounding each. The similar problemfor efficient packings of spheres in space also leads to highsymmetry. But there is almost nothing known about precisely why, ingeneral, efficiency leads to symmetry, and what kinds of symmetry arepossible. Twenty years ago a new metallic alloy was discovered, aphysical solution to a closely related optimization problem, and thealloy was found to possess a symmetry the nature of which is much lessobvious, or, put another way, in which the mathematics of the symmetryis less well developed. This proposal concerns the study of thesymmetries of efficient arrangements in space of spheres and polyhedraand in particular the development of a mathematical formalism in whichsuch symmetries can be usefully analyzed. Particular questions aboutthe nature of efficient packings of spheres are also specified.