Möbius kaleidocycles: a new class of everting ring linkages
Eliot Fried (Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University)
Abstract: Many of Escher’s works have become mainstays of popular culture. Famous examples include his Möbius bands and kaleidocycles. Made from six identical regular tetrahedra joined by revolute hinges, a kaleidocycle possesses a single internal degree-of-freedom that is manifested by a cyclic everting motion that brings different faces of the tetrahedra into view. We will describe "Möbius Kaleidocycles," a previously undiscovered class ring linkages made from seven or more identical links joined by revolute hinges. For each number of links, there exists a specific twist angle between neighboring hinges for which the associated Möbius Kaleidocycle possesses only a single internal degree-of-freedom, allowing for cyclic eversion, and the hinge orientations induce a nonorientable topology equivalent to that of a 3π twist Möbius band. Apart from technological applications, including perhaps the design of new organic ring molecules with peculiar electronic properties, Möbius kaleidocycles generate a myriad of intriguing questions in geometry and topology, some of which will be addressed in this talk. This is joint work with postdoctoral scholar Johannes Schönke.