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)
• Rigidity of Polyhedra web pages (McGill University - Montréal, illustrated by J.Shum)
• Les polyèdres 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
||convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects||October 2007 |