```Date: Mar 1, 2013 7:58 AM
Author: dan.ms.chaos@gmail.com
Subject: Re: Deformable platonic "solids"

On Feb 28, 8:44 pm, david petry <david_lawrence_pe...@yahoo.com>wrote:> 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_pe...@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.>> Examples>> 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)This is true  , but the situation is more subtle than that . If youhave a 'portion of the solid' that is rigid , adding furtherconnections 'withing that portion' will not reduce the global numberof degrees of freedom .Consider a cube connected by a bar to an octahedron . If we add anedge connecting two opposite vertices of the octahedron , the wholesystem will have the same number of degrees of freedom . In order tocalculate the correct number of degrees of freedom we need toeliminate 'redundant edges' . But it  helps that the Platonic solidshave a high degree of symmetry , and appear to have no 'redundantedges' (edges who's presence of absence does not affect the number ofdegrees of freedom , or  corresponds to less degrees of freedom thanexpected ) . Therefore, your deduction is correct.
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