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From: Dave Bird---St Hippo of Augustine <dave@xemu.demon.co.uk>
Subject: Re: The Platonic Solids
Date: 1997/06/30
Message-ID: <W3wRD8BcSCuzEwkp@xemu.demon.co.uk>#1/1
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References: <33B536E5.D08@math.fsu.edu>
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In article <33B536E5.D08@math.fsu.edu>, 
Michael Naylor <mnaylor@math.fsu.edu> writes:

This is a very nice little project. Is there a formula describing 
the number of faces, edges, and corners? One way to look at
the lefthand column is:  take an N-sided regular polygon,
and N more such depending from each of its outside edges.
Cutting this out, you can fold them down until they meet....
N=3 : in a point
N=4 : in a similar N-sided face
N=5 : an identical object placed below & rotated 1/(2N) of a circle
N=6 : no figure possible because there is no gap between the
  six hexagons to be folded away.
With a little patience, compasses, and a circular protractor,
it is therefore not that hard to mark out a sheet of card
so you can cut-out-and-fold into a dodecahedron.

In the righthand column you swap verteces <-> faces i.e. start
at a point (instead of an N-sided face) and conect it with N triangles
to N surrounding points (instead on N similar N-sided faces).

>
>                         The Platonic Solids
>
>                          by Michael Naylor
>
>                   (best viewed in Monaco 9 point)
>
>                                  .
>                                 /!;,
>                                / `;;,
>                               /   !;;,
>                              /    `;;;;,
>                             /      !;;;;,
>                            /       `;;;;;)
>                           /         !;;;/
>                           ``--__    `:;/
>                                 ``--_!/
>
>                             Tetrahedron
> .___________,                                          ,;/\`_
> |\'.'.'.'.'.'\                                       ,;;/  \'`_
> |:\.'.'.'.'.'.\                                    ,;;;/    \.'`_
> |::\'.'.'.'.'.'\                                  ;;;;/      \'.'!
> |:::+-----------+   Hexahedron        Octahedron  ;;;/        \.'!
> |:::|           |                                 ;;/          \'!
> |:::|           |                                 :/____________\;
> \:::|           |                                 `-_          _-
>  \::|           |                                   `-_      _-'
>   \:|           |                                     `-_  _-'
>    \!___________!                                        '-'
>
>                     Dodecahedron     Icosahedron
>
>            .----------,,
>          ,"__         _-:,                         ._.
>         /    ""--_--""...:\                    ,_-','.`-_
>        /         |.........\               ,_-'  ,'   `. `-_,
>       /          |..........\             !' `-,-_______'-'::
>      /,         _'_........./:            ;   / \      /.\:::
>      ! -,    _-"   "-_... ,;;:            !  /   \    /...\::
>      \   -_-"         "-_/;;;;            ! /     \  /.....\:
>       \   \             /;;;;'            ;'--.._..\/_._..-;;
>        \   \           /;;;;               '-_     :!;;;;;;'
>         '.  \         /;;;'                   `-_, :!;;;'
>           "-_\_______/;;'                         `-!'           mn


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