Jessen's orthogonal icosahedron

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.
 
This polyhedron 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 key F 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.
 
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.
Paul Mason discovered an other shaky polyhedron: five square pyramids and a square antiprism assembled on a wire cube.

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)

other flexible polyhedra

  • Bricard's octahedra: assemblings of two "square pyramids", they have intersecting faces and thus can only be realized as articulated structures with twelve "edges"
  • Connellys's "sphere" uses the Bricard's idea to avoid intersecting faces
  • Steffen's polyhedron: with 9 vertices and 14 triangular faces it is the simplest
 
references:   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)


  summary   November 2003
updated 05-08-2013