This story started with our desire to expand the capability of the Geometiles® to make transformable hinged constructions. In particular, we were interested in making flexagons using the equilateral and square Geometiles tiles. Flexagons are polygons typically made of paper, which can be folded to expose certain faces and hide others. They were invented by Arthur Stone in 1939, and he formed a “Flexagon Committee” along with Richard Feynman, John Tukey and Bryant Tuckerman to further explore their properties. Flexagons were later popularized by Martin Gardner, and they still mesmerize children and adults alike. The rich and varied history of flexagons and related transformable structures, some of which is described in this article, makes them a good subject for a future interactive workshop or exhibit.
It seemed that all we needed was a small spacer with connections that mates with the existing pieces. This turned out to be largely correct. The spacer needed to be topologically equivalent to a Moebius strip (rather than a cylinder) in order to make the hexaflexagon possible. It turns out that the Moebius-like spacers, which we call Mobies, provided a much richer collection of structures than the hexaflexagon and tetraflexagon for which they were originally designed. We can now do a nice demonstration of ”splitting” the Moebius strip.
Most notably, we were able to build hinged rings of polyhedra which admit cyclical rotations. One was the infinity cube, and the other, a rotating ring of 12 square antiprisms. The latter is a modification of the rotating ring of 14 gyroelongated square bipyramids found on page 94 of the book Mathematical Tapestry by Peter Hinton and Jean Pedersen.
This brings us to the natural question: what other constructions can be made possible with the Mobies? We invite you to explore the usefulness of the system for modeling of other types of structures. We are eager to see how this story evolves!
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