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下，研究一般维欧氏空间和双曲空间的球面或多面体最优稠密填充的对称性。这类填充问题很少被解决，但我们的目标更多地是将这类问题作为一类来研究，以使它们是适定的，具有合理的存在和唯一性条件，特别是研究解的对称性。这项工作的部分动机是发现了非周期性的瓷砖，例如风筝和省道多边形在飞机上的彭罗斯瓷砖，这种瓷砖可以理解为此类布局问题的解决方案。长期以来，非分划的对称性一直与遍历理论的数学联系在一起，使用被平铺空间的平移群或全同余群作为动力学。因此，Ttilings的对称性与结合动力系统的共轭类有关。本文对布局问题的对称性提出了几个问题，有的针对一般的定性行为，有的针对重要的特殊情况。例如，证明了固定半径球面对双曲空间最稠密填充的对称性对于不同的半径是不同的。(已经知道，对于大多数半径，最密集的堆积是非周期的--它们不可能具有晶体对称性。)从更一般的意义上说，在欧几里得或双曲空间中，“一般”的包装问题只有非结晶学的最优解。任何试图把尽可能多的便士挤到桌面上的人都已经看到，最有效的排列也是非常对称的，每个周围环绕着6便士。在太空中对球体进行有效包装的类似问题也会导致高度对称性。但是，关于效率究竟为什么会导致对称性，以及什么样的对称性是可能的，我们几乎一无所知。二十年前，一种新的金属合金被发现，这是一个密切相关的最优化问题的形而上的解，这种合金被发现具有对称性，其性质非常不明显，或者换句话说，对称性的数学不太发达。这项建议涉及研究球和多面体空间中有效排列的对称性，特别是发展一种数学形式主义，在这种形式中可以有效地分析这种对称性。关于球的有效堆积的性质的特殊问题也被详细说明。

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.