On Thursday, February 28, 2013 4:48:04 AM UTC-8, Richard Tobin wrote: > In article <firstname.lastname@example.org>, > david petry <email@example.com> wrote: > >Any line segment joining two points reduces the > >total number of degrees of freedom of the system of points and line > >segments by one.
> Not in general. Consider a deformable solid with a square face. > Joining two opposite corners will make that square rigid. Joining > the other two will have no further effect, while adding a line > somewhere else in the solid may.
Yes, of course, if the degrees of freedom are already minimal, they can't be reduced further. But the answer I gave does show us how to answer the original question.
"THEOREM" If a platonic solid has V vertices and E edges, then it will be rigid in the sense of Frederick Williams if and only if 3V - E = 6.
Tetrahedron: V = 4, E = 6, 3V-E = 6 (rigid) Octahedron: V = 6, E = 12, 3V-E = 6 (rigid) Cube: V = 8, E = 12, 3V-E = 12 (not rigid) Dodecahedron: V = 20, E = 30 3V-E = 30 (not rigid) Icosahedron: V = 12, E = 30, 3V-E = 6 (rigid)