Continuous Flattening of Non-convex Quadrangular Pyramids with Every Edge Rigid

Abstract

Continuous flattening of the surface of a given polyhedron is folding through a continuous process without tearing and stretching the surface into a multi-layered flat-folded state with a finite number of creases. It is known in the literature that the surface of any convex polyhedron can be continuously flattened by folding some edges and faces. This paper focuses on a non-convex polyhedron with a continuous flattening motion that keeps every edge rigid. We call such flattening the edge-rigid-flattening. We discuss a lower bound on the number of edges of polyhedra with the edge-rigid-flattening, and we show that a given polyhedron with the edge-rigid-flattening attains the lower bound if and only if the polyhedron is a non-convex quadrangular pyramid satisfying the local flat-foldability condition at the apex.