I am aware of two other movies of the Steffan polyderon that one can find online. The one on Wolfram Math World is a rather crudely done animation. There is a movie of a paper or cardboard model on YouTube.



You will find explanations of why this is significant if you search the web. Here is a brief explanation.



Flexibility and rigidity are intuitive concepts. Imagine putting together a tinker-toy structure, but with ideal joints so the rods are completely free to pivot. In spite of that freedom, a triangle so constructed is obviously rigid, but a square is not; it doesn't have enough rods, so we say that it is generically flexible. That kind of flexibility is too easy and not very interesting, so we use only triangles.



Now imagine building a geodesic-dome-like structure out of triangles such that every joint where triangles meet is completely flexible (the model used here has duct tape for joints). Cauchy proved in 1812 that if it is convex, it must be rigid - in spite of the flexibility of all the joints.



After that proof, people came to believe that there were no flexible polyhedra at all. But in 1897 R. Bricard discovered three nonconvex octahedra that are indeed flexible, but at a price. As the dihedral angles vary continuously, the faces intercross, so these octahedra cannot actually exist as authentic objects in real space. They resemble some of the unreal objects of M. C. Escher. An example of a real flexible polyhedron, long thought to be impossible, was given in 1977 by Robert Connelly. This amazing polyhedron is non-generically flexible: it seems to have enough rods (constraints) to be rigid but it is not. Nonetheless, Bricard's approach remains useful.



The model here of Steffen's construction was built by Robert Uhrlass, a student at Fordham University in April 2010. It is made of aluminum and duct tape. Thanks Rob! Making such a model is not easy. If you do it out of paper or cardboard, the material is too flimsy, bends easily, and is not convincing; it ends up looking like origami. If you use thick cardboard, then it is not realistic, the tolerances are small, and you will have a lot of trouble getting it together. Solving the problem of the joints is not easy either.



It's great to have a model of aluminum that one can not only move gingerly, but squeeze.



Feel free to download the movies. On my computer, at least, the downloaded mp4 files look better than they do when a browser plays them.



-- Prof. Robert H. Lewis, Fordham University

http://fordham.academia.edu/RobertLewis



movie 1



movie 2



movie 3









