# 3-coloring a truncated icosahedron **URL:** \ **Category:** Factual Questions\ **Created:** [July 12, 2002, 10:46pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707 "2002-07-12T22:46:16Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![bryanmcc](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@bryanmcc](https://boards.straightdope.com/u/bryanmcc)\ **Post date:** [July 12, 2002, 10:46pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/1 "2002-07-12T22:46:16Z")
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](http://web.merrimack.edu/~thull/phzig/phzig.html) 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](http://math.ucsd.edu/~llu/dgea/). Any help, hints, links, etc. would be apprectiated. Thanks! -b ---
**Author:** ![bryanmcc](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@bryanmcc](https://boards.straightdope.com/u/bryanmcc)\ **Post date:** [July 13, 2002, 2:30pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/2 "2002-07-13T14:30:15Z")
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](http://babysmurf.er.usgs.gov/~bjm/Picture_28.bmp) -b ---
**Author:** ![Fear\_Itself](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/fear_itself/32/19637_2.png) [@Fear\_Itself](https://boards.straightdope.com/u/Fear_Itself)\ **Post date:** [July 13, 2002, 2:38pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/3 "2002-07-13T14:38:45Z")
Don’t know if this helps, but it looks cool: [truncated icosahedron](http://plaza13.mbn.or.jp/~jinsho/polyhedra/LG3D/TcIcosa.htm) ---
**Author:** ![bryanmcc](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@bryanmcc](https://boards.straightdope.com/u/bryanmcc)\ **Post date:** [July 13, 2002, 4:03pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/4 "2002-07-13T16:03:21Z")
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](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 ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [July 13, 2002, 9:17pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/5 "2002-07-13T21:17:55Z")
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? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 13, 2002, 9:43pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/6 "2002-07-13T21:43:29Z")
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. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [July 13, 2002, 10:08pm UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/7 "2002-07-13T22:08:14Z")
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. 😃 [Of course, I did. This place has a ton of Platonic and Archimedian solids, all rotatable via Javascript. Enable popups for this one, kids. It’s worth it.](http://plaza13.mbn.or.jp/~jinsho/polyhedra/index.html) 🙂 (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.) ---
**Author:** ![bryanmcc](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@bryanmcc](https://boards.straightdope.com/u/bryanmcc)\ **Post date:** [July 14, 2002, 1:31am UTC](https://boards.straightdope.com/t/3-coloring-a-truncated-icosahedron/118707/8 "2002-07-14T01:31:06Z")
**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