|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 bases of the isosceles triangles are the great sides of three rectangles two by two orthogonal and with sides' ratio 2.
It can be very lightly deformed by acting on the angles of the pairs of the isosceles triangles.
All its dihedral angles are right (adjacent faces are orthogonal).
The centres of the eight remaining equilateral faces are the vertices of a cube (use the F key to see it).
Beware! the convex envelope is a non regular icosahedron sometimes called pseudo-icosahedron; it also has six pairs of isosceles faces and only three planes of symmetry.
Remark: 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 polyèdres 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)
• How "shaky" is the Jessen's orthogonal icosahedron?
|November 2003 |