The Platonic Solids

In article <33B...@math.fsu.edu>, 
Michael Naylor <mna...@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


       @@@                                          @@@    
      @("}                                          {")@
     @(.(.)\__                                  __/(.).)@
    @@@) (    """"""""""""""""""""""""""""""""""    ) (@@@
      ( . )            dav...@xemu.demon.co.uk       ( . )
  \    )  /      http://www.xemu.demon.co.uk        \  (    /
  \\.-' ,'                 /index.html               ', `-_// 
  // ~-~   """"""""""""""""""""""""""""""""""""""""""   '~ \\
  /                                                         \
   "Consciousness is a product of the brain no more & no less
    than earwax is a product of the ear"-Martin Hunt on a.r.s 

Shimrod (csg...@wing.rug.nl) wrote:
>> 
>>                      Dodecahedron     Icosahedron
>> 
>>             .----------,,
>>           ,"__         _-:,                         ._.
>>          /    ""--_--""...:\                    ,_-','.`-_
>>         /         |.........\               ,_-'  ,'   `. `-_,
>>        /          |..........\             !' `-,-_______'-'::
>>       /,         _'_........./:            ;   / \      /.\:::
>>       ! -,    _-"   "-_... ,;;:            !  /   \    /...\::
>>       \   -_-"         "-_/;;;;            ! /     \  /.....\:
>>        \   \             /;;;;'            ;'--.._..\/_._..-;;
>>         \   \           /;;;;               '-_     :!;;;;;;'
>>          '.  \         /;;;'                   `-_, :!;;;'
>>            "-_\_______/;;'    mn                   `-!'           mn
>> 


My Aunty used to have one of those "dodecahedrons"...


             .----------,,
           ,"__:::::::: _-:,                                         
          /....""--_--""...:\                                        
         /.........|.........\                                       
        /... 12 ...|.... 4 ...\                                      
       /.........._'_........./|                                     
       | -,... _-"..."-_... ,..|                                     
       \ 1.-_-"........."-_/.6.|                                     
        \...\..... 8 .... /.../'                                     
         \...\.........../.../                                       
          '. .\........ /../'                                        
            "-_\_______/_/'                                          
                                                                     

It was made of crystal glass and had numbers on each facet.

You rolled the "multihedron", which had many more faces than a 
dodecahedron, and Aunty had a secret list of the numbers. Each 
number on the list told part of your fortune. 

Easier than reading tea leaves !

Bye,