标题： 从高度优化的多面体通道中获得的一些新见解 显示英文标题

标题： Some New Insights from Highly Optimized Polyhedral Passages

摘要： 如果一个形状可以被切割出一个孔，使得第二个相同的形状能够直线穿过第一个形状的内部，则该形状具有鲁珀特特性。 证明立方体具有鲁珀特特性首次在300多年前被展示。 目前尚不清楚三维空间中的每个多面体是否都是鲁珀特的。 我们提出了一种定制的次梯度方法，为给定的多面体提供高质量的局部数值优化。 通过广泛的数值搜索，我们改进了超过一半的柏拉图立体、阿基米德立体和卡塔兰立体以及众多约翰逊立体的最佳已知通道。 我们的高精度求解支持了一个简单的形式的新猜想，即四面体的最佳通道。 尽管进行了计算搜索，仍有三个阿基米德立体和两个卡塔兰立体尚未解决，这进一步提供了反对所有多面体都是鲁珀特的猜想的负面证据。 显示英文摘要

摘要： A shape possesses Rupert's property if a hole can be cut through it such that a second identical copy of the shape can cleanly pass straight through the interior of the first. Such a passage proving cubes are Rupert was first shown more than 300 years ago. It remains open whether every polyhedron in three dimensions is Rupert. We propose a customized subgradient method providing high-accuracy local numerical optimization of the quality of a passage for a given polyhedron. From extensive numerical searches, we improve these best-known passages for more than half of the Platonic, Archimedean, and Catalan solids and for numerous Johnson solids. Our high accuracy solves support a new conjecture of a simple form for the Tetrahedron's optimal passage. Despite our computational search, three Archimedean and two Catalan solids remain open, providing further negative evidence against the conjecture that all polyhedrons are Rupert.