Mobility of a class of perforated polyhedra

A class of over-braced but typically flexible body-hinge frameworks is described. They are based on polyhedra with rigid faces where an independent subset of faces has been replaced by a set of holes. The contact polyhedron C describing the bodies (vertices of C) and their connecting joints (edges of C) is derived by subdivision of the edges of an underlying cubic polyhedron. Symmetry calculations detect flexibility not revealed by counting alone. A generic symmetry-extended version of the Grübler–Kutzbach mobility counting rule accounts for the net mobilities of infinite families of this type (based on subdivisions of prisms, wedges, barrels, and some general inflations of a parent polyhedron). The prisms with all faces even and all barrels are found to generate flexible perforated polyhedra under the subdivision construction.

The investigation was inspired by a question raised by Walter Whiteley about a perforated polyhedron with a unique mechanism reducing octahedral to tetrahedral symmetry. It turns out that the perforated polyhedron with highest (Oh) point-group symmetry based on subdivision of the cube is mechanically equivalent to the Hoberman Switch-Pitch toy. Both objects exhibit an exactly similar mechanism that preserves Td subgroup symmetry over a finite range; this mechanism survives in two variants suggested by Bob Connelly and Barbara Heys that have the same contact graph, but lower initial maximum symmetry.