Updates to Turnabout 5, "Piano-Hinged Polyhedra",
in Hinged Dissections: Swinging & Twisting, by Greg N. Frederickson

Discussion by Peter Cromwell

As I wrote in the afterword, Peter Cromwell's Polyhedra gives a detailed discussion of the rigidity and flexibility of polyhedra. His Chapter 6 contains a wonderful description of how Robert Connelly came to modify Bricard's construction so that no two faces would interpenetrate.

Hinging a small stellated dodecahedron ?

Izidor Hafner, in the Department of Mathematics at the University of Ljubljana in Slovenia, posted an intriguing video (5.2 MBytes) on his website. The video shows the flexing of a nonconvex solid of 60 isosceles triangular faces that, when in the most symmetric configuration, looks a lot like the small stellated dodecahedron {5/2,5}, which is based on 12 pentagrams. Each isosceles triangle is hinged to three other triangles by piano hinges along its edges. Unlike small stellated dodecahedron, which has its visible portions being isosceles triangles with edges in the ratio of approximately 1.618, the solid in the video has isosceles faces with edges in the ratio of 2. Izidor says that more likely that his solid is only infinitesimally flexible.

Copyright 2014, Greg N. Frederickson.
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Last updated June 13, 2014.