|This shaky polyhedron - infinitesimally flexible - may be constructed by replacing six pairs of equilateral triangles in a regular icosahedron with pairs of isosceles triangles (the common bases are the great sides of the three golden rectangles defined by the vertices). It can be very lightly deformed by acting on the angles of the pairs of the isosceles triangles.
The centres of the eight remaining equilateral faces are the vertices of a cube (use the key F to see it).
If this icosahedron is constructed with paper (non rigid faces) then the isosceles triangles may be folded inside by pairs to get a regular octahedron.
Two important results concerning the flexible polyhedra:
• A convex polyhedron is rigid. (Cauchy's rigidity theorem, 1813)
• During the deformation of a flexible polyhedron its volume remains constant. (bellow's conjecture, Connelly-Sabitov-Walz, 1997)
The Penguin Dictionary of Curious and Interesting Geometry by David Wells (Penguin, London - 1991, page 161)
Les polyŤdres flexibles et la conjecture du soufflet by Thierry Lambre (bulletin 471 APMEP, page 533, in French)
The Bellows Conjecture by Ian Stewart
Polyhedra by Peter R. Cromwell (Cambridge University Press - 1997, page 239-246)