Continuous Flattening of Multi-layered Pyramids with Rigid Radial Edges

Abstract

Several methods described in the literature have proved that any convex pyramid can be continuously flattened. Recently, the problem of continuous flattening of polyhedra having divisions, i.e., polyhedra in which some of the edges are incident to three or more faces, has been proposed. However, for such multi-layered structures, continuous flattening motions are unknown. In this study, under the assumption that every radial edge is rigid, we prove that a continuous flattening motion exists for a pyramid with a convex base. Moreover, in a similar manner, we demonstrate that a continuous flattening motion exists for a multi-layered pyramid having a common convex base, with each apex having a common perpendicular foot. Finally, we illustrate an example of a multi-layered pyramid with a non-convex base that cannot be continuously flattened while maintaining the rigidity of the radial edges.