Some New Insights from Highly Optimized Polyhedral Passages

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 polyhedra are Rupert.