The Symmetry of Densest Packings of Space

这个建议是在希尔伯特的第十八个问题的精神，研究对称的最佳密集包装，球或多面体，欧几里德和双曲空间的一般维数。很少有特殊的包装问题已经解决，但我们的目标是更直接地研究这样的问题作为一个类，指定他们，使他们是适定的，合理的条件存在和唯一性，and in particular具体to thesymometries学习of the solutions解决方案.这项工作的动机部分是由于发现了非周期性的平铺，如彭罗斯平铺的飞机的风筝和飞镖多边形，平铺可以理解为解决这样的包装问题。代数拼接的对称性一直与遍历理论的数学联系在一起，使用平移群或空间的全同余群作为动力学。的对称性的tilings，然后有关的共轭类的associateddynamic系统。这里提出了几个问题的对称性包装问题，一些针对一般的定性行为和一些针对重要的特殊情况。例如，它建议表明，对称的双曲空间的密填充的球的固定半径是不同的，为不同的tradii。(It已经知道，对于大多数半径，密度填充是非周期性的--它们不可能具有晶体对称性。更一般地说，它是为了证明一个“一般的”包装问题，在欧几里得或双曲空间，只有最佳的解决方案，而不是结晶。任何人谁试图挤压尽可能多的便士到一个桌面上已经看到，最有效的安排也是非常对称的，有六个便士周围。空间中球体有效填充的类似问题也导致了高度对称性。但是对于为什么一般来说效率会导致对称性，以及什么样的对称性是可能的，我们几乎一无所知。20年前，人们发现了一种新的金属合金，它是一个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却不那么明显，或者换句话说，这种对称性的数学还没有得到很好的发展。这一建议涉及到球面和多面体空间中有效排列的对称性的研究，特别是一种数学形式的发展，在这种形式中，这种对称性可以被有效地分析。特别的问题aboutthe性质的有效包装的领域也被指定。

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.