Rupert property of some particular n-simplices and n-octahedrons

Abstract

Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in \(\mathbb {R}^3\). For higher dimensions, the n-dimensional cube and the regular n-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in n dimensions. In particular, we show that some particular n-dimensional simplices, generalized n-dimensional octahedrons and some related solids in \(\mathbb {R}^n\) have the Rupert property using arbitrarily small rotations and translations.