Platonic固体和Archimedean固体的稠密填充
Dense Packings of the Platonic and Archimedean Solids
摘要：
致密堆积已成为研究液体、玻璃态和晶体态物质、颗粒介质、非均质材料和生物系统结构的有用模型。探索离散填充的对称性和其他数学性质是离散几何和数论中一个长期感兴趣的问题。以前的工作主要集中在球形颗粒，而对致密多面体填料知之甚少。我们制定了一个自适应的基本细胞周期性边界条件的多面体内产生密集包装的问题作为一个优化问题，我们称之为自适应收缩细胞（ASC）计划。这个新的优化问题在这里解决（使用各种多粒子初始配置），以找到在三维欧氏空间中的每个柏拉图固体的稠密包装。我们找到了密度分别为$0.782...$，$0.947...$，$0.904...$，和$0.836...$的四面体，八面体，十二面体和二十面体的密集的已知填充.不同于denitrifetrahedral packing，这必须是一个非布拉维格包装，denitrifetrahedral packing的其他非平铺柏拉图固体，我们得到的denitrifetrahedral packing是他们以前已知的最佳（布拉维）格包装。我们的模拟结果，严格的上限，我们推导出，和理论论证导致我们强烈的猜想，即中心对称的柏拉图和阿基米德固体的dennial包装给出了相应的dennial晶格包装。这是开普勒对这些固体的球面猜想的类比。
英文摘要：
Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of long-standing interest in discrete geometry and number theory. 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 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 novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analog of Kepler's sphere conjecture for these solids.
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