Reversing a polyhedral surface by origami-deformation

AbstractWe introduce a new variety of flexatube, a rhombotube. It is obtained from a cardboard rhombohedron consisting of six rhombi with interior angles 60∘ and 120∘, by removing a pair of opposite faces, 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 rhombotube 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 the 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