Simplices passing through a hole
The authors mention first a few simple and known observations regarding a possibility to push a regular tetrahedron \(\sigma^3\) of edge-length 1 through a circular hole \(H\) in a plane \(P \subset {\mathbb R}^3\). In particular, \(\sigma^3\) can pass through \(H\) if diam \(H \geq 2/\sqrt{3}\) and cannot if diam \(H< \sqrt{3}/2\) (which is the width of a face of \(\sigma^3\)). Then they extend the problem to an arbitrary convex hole \(H\) in a hyperplane \(P\) in 3-, 4-, and 5-dimensional Euclidean space. They prove the following results. Theorem A. There exists a convex hole \(H \subset P\) of diameter \(\frac{\sqrt{3}}{2}\) and width \(\frac{\sqrt{2}}{2}\) such that the regular tetrahedron \(\sigma^3\) moving in \({\mathbb R}^3\) can pass through \(H\). Theorem B. There is a convex hole \(H \subset P\) of diameter \(\frac{\sqrt{3}}{2}\) and width less than \(\frac{\sqrt{2}}{2}\) such that the regular simplex \(\sigma^4\) moving in \({\mathbb R}^4\) can pass through \(H\). Theorem C. There is a convex hole \(H \subset P\) of diameter \(\frac{\sqrt{3}}{2}\) and width \(\frac{\sqrt{6}}{3}\) such that the regular simplex \(\sigma^5\) moving in \({\mathbb R}^5\) can pass through \(H\).
scientific article; zbMATH DE number 5000045