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Topic: Deformable platonic "solids"
Replies: 23   Last Post: Mar 12, 2013 8:11 PM

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 David Bernier Posts: 3,892 Registered: 12/13/04
Re: Deformable platonic "solids"
Posted: Feb 28, 2013 2:26 AM

On 02/28/2013 01:27 AM, Dan wrote:
> On Feb 27, 9:21 pm, Frederick Williams <freddywilli...@btinternet.com>
> wrote:

>> Suppose the platonic solids aren't solid at all but are made of rigid
>> line segments with completely flexible hinges at the vertices. The cube
>> can be flattened into a... um... non cube. The tetrahedron, octahedron
>> and icosahedron cannot be deformed at all. But what about the
>> dodecahedron, can it be deformed?
>> --
>> When a true genius appears in the world, you may know him by
>> this sign, that the dunces are all in confederacy against him.
>> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

>
> The question can be formalized in the following manner :
> Define a 'semi-Platonic' solid as a solid with equal number of edges
> per face , same number of faces as a regular counterpart , all edges
> of the same length .
> It's the same as a Platonic solid , just drop the condition of equal
> angles per face , and the 'planar' nature of faces .
>
> The question is : Is a 'semi-Platonic' solid necessarily platonic?
> That means , for a Platonic solid , does there exist a solid with the
> same properties except different angles?
> As pointed out , solids with triangular faces are not deformable .
> (being an equilateral triangle uniquely determines angles , as opposed
> to having a higher number of sides)
>
> As calculated , all Platonic solids with non-triangular faces are
> deformable .
>

Are all agreed that the dodecahedron is deformable?
I'm myself not sure what to think at present.

In the literature, where solid faces don't count and
only the linkages between vertices as though by wiring matter,
the terminology:
"infinitesimal rigidity of frameworks"

seems standard and perhaps the most relevant.

dave
--
dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.