Markus Becker wrote:
> 
> Remco de Korte wrote:
> >
> > Is that so? I'm willing to believe it, but I wonder if there's some way to prove
> > that or make it acceptable for a mathematician (which I'm most definitely not).
> 
> Yes, this _is_ so. It is mathematically proven. A simplified (and short)
> explanation is that you can only tile certain surfaces with certain
> patches.
> The certain surface bein the spere and the patches being triangles,
> pentagons
> ans so on for the sphere surface.
> If you have access to Scientific American, you should be able to find
> articles about it.
> 
> For some reading, look here:
> http://www.rose-hulman.edu/~brought/Epubs/REU/Wabash.html
> 
> Markus

Aha! Thanks! This means I won't have to feel frustrated because I can't solve
this little problem.

CU!

Remco

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I don't know why I didn't see this thread earlier... it's already one
month old...

However, I just recently wrote and published a macro for geodesic
spheres which happen to be bodies that are constructed by dividing the
surface of a sphere into triangles. The principle behind it starts from
the three basic polyhedra with triangular faces:
  tetrahedron  4 faces
  octahedron   8 faces
  icosahedron  20 faces
and divides each triangular face into smaller triangles. Though it is
NOT possible to put any number of points (or triangles/corners) onto a
sphere you can choose from quite a variety of values by just selecting
the basic polyhedron and the "frequency" of subdivision for each face.
The number of faces on the final sphere is then calculated as 

  number of faces on polyhedron * (freq^2)

so you can get the following values:

freq      tetrahedron    octahedron   icosahedron
  1             4             8            20
  2            16            32            80
  3            36            72           180
  4            64           128           320
  5           100           200           500
  6           144           288           720
  8           256           512          1280
etc.    

By defining your own "drawing" (better: object placing) routine for each
face, edge and corner, you can easily distribute whatever object you
want on the faces.

You can download the macro, documentation  and see some examples at
http://www.geocities.com/SiliconValley/Lakes/5432/povray/geodesic.html


Uwe.

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