Steffen's flexible polyhedron

In 1766 Euler proposed the "rigidity conjecture" and Cauchy proved it for convex polyhedra in 1813.
Klaus Steffen constructed a symmetric flexible Connelly polyhedral sphere. It's worth to build your own model of this curious object which is the simplest flexible polyhedron (14 triangular faces and 9 vertices).

net

net by Peter R. Cromwell

The LiveGraphics3D applet has some difficulties to well display faces which are almost coplanar.

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)

  • Bricard's octahedra (1897): assemblings of two "square pyramids", they have intersecting faces and thus can only be realized as articulated structures (Raoul Bricard was a French engineer).
  • Robert Connelly's "sphere" (1978) uses the Bricard's idea to avoid intersecting faces (a model simplified by Kuiper and Deligne has 18 faces and 11 vertices).
  • Jessen's orthogonal icosahedron is infinitesimally flexible (shaky polyhedron)

references: •  Rigidity of Polyhedra  web pages (McGill University - Montr√©al, illustrated by J.Shum)
•  Les poly√®dres flexibles et la conjecture du soufflet  by Thierry Lambre (bulletin 471 APMEP, page 533, in French)
•  Polyhedra  by Peter R. Cromwell (Cambridge University Press - 1997, pages 239-246)
•  Steffen's polyhedron on MathWorld and on mathematik.com


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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects October 2007
updated 27-10-2007