```Date: Feb 28, 2013 1:44 PM
Author: David Petry
Subject: Re: Deformable platonic "solids"

On Thursday, February 28, 2013 4:48:04 AM UTC-8, Richard Tobin wrote:> In article <511e7643-79fb-4418-9108-16b317c87dff@googlegroups.com>,> david petry  <david_lawrence_petry@yahoo.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.ExamplesTetrahedron: 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)
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