Lonely Solids

Abstract

A three-dimensional solid has the Rupert property if a congruent copy of the solid can pass through a hole cut through it without splitting it. We extend this idea to pairs of convex solids: two solids are called \textit{friends} if each can pass through a suitable hole in the other. A solid is called \textit{lonely} if it has no friends, including itself. We show that a convex solid is lonely if and only if it has constant width. We also show that every convex solid that does not have constant width has a particularly simple friend: an arbitrarily long and arbitrarily thin rectangular cuboid. Finally, we prove that all non-constant-width convex solids lie in a single connected component of the friendship graph. More precisely, any two such solids are connected by a chain of at most ``three handshakes''.