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3-Colouring for Truncated Icosahedron

23 September 2009 - 10:59pm — Sara
Related image: 
Truncated Icosahedron from PHiZZ Units (Tom Hull)
3-Colouring for Truncated Icosahedron

Today someone asked whether you could colour a truncated icosahedron with 3 colours only, such that no two adjacent edges had the same colour.
After some research I found that it was possible, and that one way of finding such a colouring is as follows:

1. Draw the corresponding graph.
2. Find a Hamiltonian circuit in that graph. [Got this from Tom Hull's submission to Origami 3]
3. Colour the Hamiltonian circuit with two colours, alternating between them as you go along the path.
4. Give the remaining edges the third colour.

This picture shows the result I got. It's by no means the only colouring that is possible, but the colours are relatively evenly distributed on it. That is, every pentagon and hexagon has at least one module of each of the 3 colours. This might be interesting to those of you out there that do lots of modular origami.

Comments

12 April 2011 - 1:18pm — Kazza the Blank One (not verified)

Path doesn't join?

I was just staring at this and I think there might be a mistake .. I think the Hamiltonian path is meant to be one continuous line? It looks like you have two separate paths there?

17 April 2011 - 8:47am — Sara

Ah, read carefully

Ah, read carefully:

1. Draw the corresponding graph.
2. Find a Hamiltonian circuit in that graph. [Got this from Tom Hull's submission to Origami 3]
3. Colour the Hamiltonian circuit with two colours, alternating between them as you go along the path.
4. Give the remaining edges the third colour.

This says that once you have a Hamiltonian circuit, you'll indeed chunk it up and color it with two different colors. :)

-- Sara

17 April 2011 - 1:10pm — Kazza the Blank One (not verified)

Looks better now!

Looks fixed now :)

I made a large phizz buckyball with seven hexagons on each of the "faces" between the pentagons. The hamiltonian path on it surprised me, but I'm yet to figure out how to show it nicely (and I don't fancy drawing a diagram like the one above with all 270 edges! heh)

http://kazza.id.au/2011/04/large-phizz-buckyball.html

1 January 2011 - 4:01am — Elvis D. (not verified)

our of curiostity does this

our of curiostity does this show all pieces(all 90)

2 January 2011 - 1:31pm — Sara

Sure!

Yes, it does.

24 September 2009 - 1:10am — Anonymous

Interesting.

I read that wikipedia page all the way through, and understood 40% of it, enough to know what you are talking about, and the concept.
WOW!!!!!!