Reversing a Polyhedral Surface

We introduce a new variety of flexatube, a rhomboflexatube. It is obtained from a cardboard rhombohedron by removing a pair of opposite faces (rhombi), and then subdividing the remaining four faces by pairs of diagonals. It is reversible, that is, it can be turned inside out by a series of folds, using edges and diagonals of the rhombi. To turn a rhomboflexatube inside out is quite a challenging puzzle. We also consider the reversibility of general polyhedral surfaces. We show that if an orientable polyhedral surface with boundary is reversible, then its genus is 0 and for every interior vertex, the sum of face angles at the vertex is at least 2π. After defining tube-attachment operation, we show that every polyhedral surface obtained from a rectangular tube by applying tube-attachment operations one after another can be subdivided so that it becomes reversible. 2000 Mathematics Subject Classification. Primary: 52C25, Secondary: 00A08, 97A20. Key word and phrases. flexatube; rhomboflexatube; reversible; subdivision-reversible.