Hinged Dissection of Polypolyhedra

摘要

This paper presents a general family of 3D hinged dissections for polypolyhedra, i.e., connected 3D solids formed by joining several rigid copies of the same polyhedron along identical faces. (Such joinings are possible only for reflectionally symmetric faces.) Each hinged dissection consists of a linear number of solid polyhedral pieces hinged along their edges to form a flexible closed chain (cycle). For each base polyhedron P and each positive integer n, a single hinged dissection has folded configurations corresponding to all possible polypolyhedra formed by joining n copies of the polyhedron P. In particular, these results settle the open problem posed in [7] about the special case of polycubes (where P is a cube) and extend analogous results from 2D. Along the way, we present hinged dissections for polyplatonics (where F is a platonic solid) that are particularly efficient: among a type of hinged dissection, they use the fewest possible pieces.