论文标题
普通多边形的密集平面组包装
Densest plane group packings of regular polygons
论文作者
论文摘要
在建模物理和生物系统以及离散和计算几何形状的背景下，已经对足够致密的常规凸多边形（$ n $ gon）包装进行了广泛的研究。以前的结果主要是关于最密集的晶格或双晶格配置。在这里，我们通过限制了二维欧几里得空间的一致副本的一般填料问题的一般填料问题的构造空间来考虑所有二维晶体学对称组（平面组）。我们将平面组的填料问题提出为非线性约束优化问题。通过大约解决此问题的熵信任区域包装算法，我们检查了所有$ 17 $平面组中各种$ n $ gons的一些已知且未知的最稠密的包装，以及对每$ n $ n $ n $ n $ n $ gon的密集飞机组包装的共同对称性的猜想。
Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various $n$-gons in all $17$ plane groups and state conjectures about common symmetries of the densest plane group packings for every $n$-gon.
