Can anyone help me properly three-color the edges of a truncated icosahedron (i.e., Buckyball or soccer ball)? I’m trying to make one out of origami with the instructions from this site. It says there that it should be possible, but I’m having a bitch of a time figuring out how. I can’t find a planar graph of a truncated icosahedra, but you can generate one here. Any help, hints, links, etc. would be apprectiated. Thanks!
-b
Here is a planar graph that I’ve generated for the truncated icosahedron that I’m trying to properly three-edge color:
http://babysmurf.er.usgs.gov/~bjm/Picture_28.bmp
-b
Don’t know if this helps, but it looks cool:
Yeeeeeess!! I think I’ve figured it out. Assuming I haven’t screwed up on some vertex and not noticed it, I think this graph is properly three-edge colored:
http://babysmurf.er.usgs.gov/~bjm/Picture_28a.bmp
So, the question now becomes is this the only way to color this shape? Is there some other coloration that would be more symetrical or attractive?
-b
That picture file is Forbidden for me, bryanmcc. Can you change the permission on it?
Also, just to make it clear, you want no adjacent shapes to be the same color, right?
I think he’s actually trying to color the edges, not the faces. Presumably, no two edges of the same color may meet at a vertex.
I just want to say that Fear Itself’s link rocks to all hell. Polyhedra you can rotate with a mouse click that are really, really quick to download (to someone sitting at the end of a dialup connection that peaks somewhere below 28.8 Kb/s).
But he didn’t traverse the directory tree to find the really nice page. 
(I have nothing to add, except that tesselation might be a good search topic. Tesselation is the art/mathematics of tiling the plane with various regular and irregular shapes, and so runs into the same problems of coloring the OP seems to have had.)
Achernar: Damn. I was afraid of that. Is there any way I can post the actual picture? Or someplace I can upload it to that would be accessible to everybody?
Chronos: Yes, you have it correct. The origami units will be the edges of the polyhedron. Imagine making the framework of a soccer ball out of toothpicks and marshmellows. Three of toothpicks will meet at each vertex, and I want none of the same colors to meet.
A “planar graph” is a way to flatten a representation of a polyhedra onto a plane. Imagine your toothpick model is made of rubber. Stretch open one pentagon, and press the whole thing onto the table so that it lies flat. This is a helpful model to try to three-edge color.
-b
