Deform the surface
©Mathematik.com 1999-2001
While a theorem of Cauchy excludes continuous deformations of convex polyhedra, this is possible for nonconvex surfaces as shown first by Connelly in 1977. The example here is due to Klaus Steffen. Challenge: can one give explicit formulas for the motion of the points during the deformation? The Povray 3.1 program which computed the surface in this page determined the points by solving a system of quadratic equations for the 27 unknowns. This happened by folding up the surface using two free parameters (see the unfolded paper model). In some sense, the geometry defines already the Groebner basis to solve the nonlinear system of equations. One parameter necessary to fit the triangles together was then determined numerically.
More information on flexible surfaces in "Proofs from the book", by M. Aigner and G.M. Ziegler, Springer 1998