Dense packings of polyhedra: Platonic and Archimedean solids.
Dense packings of polyhedra: Platonic and Archimedean solids.
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DOI：
10.1103/physreve.80.041104
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发表时间：
2009-09
期刊：
影响因子：
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通讯作者：
S. Torquato;Y. Jiao
中科院分区：
文献类型：
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作者：
S. Torquato;Y. Jiao
理解致密颗粒堆积的性质是物理、数学和生物科学中的一个热门研究课题。以往的工作主要集中在球形颗粒和很少有人知道致密的多面体填料。我们制定的问题产生密集包装的非重叠，nontiling多面体内的自适应基本细胞周期性边界条件作为一个优化问题，我们称之为自适应收缩细胞（ASC）计划。这个优化问题在这里被解决（使用各种多粒子初始配置），以找到三维欧几里得空间R3中每个柏拉图固体的稠密填充，除了立方体，它是唯一的柏拉图固体，瓷砖空间。我们发现密度为0.823的四面体、二十面体、十二面体和八面体的密度已知的堆积，0.836..., 0.904..., 0.947……分别值得注意的是，这种四面体堆积不具有长程有序性。与不一定是布拉维格填充的密排四面体填充不同，我们得到的其他非平铺柏拉图立体的密排填充是它们先前已知的最优（布拉维）格填充。我们还推导出一个简单的上界的最大密度的全等非球形粒子的包装，并将其应用到柏拉图固体，阿基米德固体，超级球和椭球。假设我们所称的“非球面性”（外圆半径与内圆半径之比）足够小，上界相对紧密，因此接近中心对称柏拉图和阿基米德固体的最佳晶格填充的相应密度。我们的模拟结果，严格的上限，和其他理论论据，使我们的猜想，即中心对称的柏拉图和阿基米德固体的dennial包装是由其相应的dennial晶格包装。这可以被认为是类似开普勒的领域猜想为这些固体。截头四面体是唯一的非手性阿基米德固体，不是中心对称的[更正]，denunciously已知的包装，其中是一个非晶格包装的密度至少高达23/24=0.958 333. .我们讨论了我们的猜想的有效性包装的超级球，棱柱，以及反棱柱，以及高维类似物的柏拉图固体。此外，我们还猜想，任何无中心对称的凸全等多面体的最优填充一般都不是格填充。最后，我们讨论了ASC方案在预测多面体纳米颗粒晶体结构和研究硬多面体随机堆积中的可能应用和推广。
Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the adaptive shrinking cell (ASC) scheme. This optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823..., 0.836..., 0.904..., and 0.947..., respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the "asphericity" (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be the analog of Kepler's sphere conjecture for these solids.The truncated tetrahedron is the only nonchiral Archimedean solid that is not centrally symmetric [corrected], the densest known packing of which is a non-lattice packing with density at least as high as 23/24=0.958 333... . We discuss the validity of our conjecture to packings of superballs, prisms, and antiprisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.