How to colour the tetrahedra in a chain of 24 to produce a rhombic dodecahedron?

September 16, 2010 3:02 PM

How should I colour each isocoles tetrahedron in a chain of 24 using three colours, so that when I fold up the chain into a rhombic dodecahedron each rhombic face is a single colour and is different to its neighbours?

I am planning to knit a rhombic dodecahedron by knitting a chain of 24 isocoles tetrahedra and folding it up. I can knit each tetrahedron in a single colour (i.e. all the faces on each single tetrahedron need to be the same), but I can change at the hinges. My question is: what sequence of colours should I use when knitting the chain so that in the finished dodecahedron each rhombic face is a single colour, and different to its neighbours?

Just thought someone might have done this sort of thing before, or might be able to figure it out without having to make a paper model and colour it in and take it apart first. If need be that's what I'll do (I have resources for that), I'm just trying to avoid the chore :-)

Also: any other colouring ideas? The technical limitations of the knitting technique I'm using mean that I can only knit stripes that are parallel to the hinges, i.e. not lengthwise in the direction of the chain.

For reference, here's a video of a kaleidocycle I made that illustrates the construction of the tetrahedral chain and the colouring possibilities.

posted by Brentusfirmus to Science & Nature (10 answers total)

Here's a Google SketchUp model of the solid. Load that into sketchup and color the faces. You can probably unwrap it in SketchUp too.
posted by snoktruix at 3:16 PM on September 16, 2010

Response by poster: Thanks for the response, but I'm not sure whether that will help because I don't just need to unfold the faces of the solid, I need to carve it up into 24 similar hinged isocoles tetrahedra that nest together (i.e. fill space) to create the dodecahedron. Or is there a way to do this in Sketchup?
posted by Brentusfirmus at 4:00 PM on September 16, 2010

Response by poster: Something extra: here is a photo of a rhombic dodecahedron created from the isocoles tetrahedra I'm talking about, from this site. Anyone know how they did it?
posted by Brentusfirmus at 4:11 PM on September 16, 2010 [1 favorite]

If you have access to Mathematica or some other CAS with a graph theory package, there should be a graph coloring function that will color the adjacency graph of the triangles in the way you need. There may also be a coloring just out there in the internets, I'd google "rhombic dodecahedron graph coloring".
posted by PMdixon at 4:15 PM on September 16, 2010

I bet you'd have more luck here
posted by meta_eli at 4:16 PM on September 16, 2010

Or, alternately, here.
posted by box at 4:34 PM on September 16, 2010

each rhombic face is a single colour and is different to its neighbours

You could 3-colour like this I think? (Which I worked out like this)

I'm not quite sure how the isocoles tetrahedrons and 24 come into what you want to do, though - you'll have to post a picture when it's done so I can understand!
posted by Mike1024 at 4:35 PM on September 16, 2010

flexagon.net has a paper model of one using 18 triangles that you could print here. There are lots of patterns there, so they may have one with 24 hiding somewhere.
posted by soelo at 7:01 PM on September 16, 2010

Oops, ignore the part about triangles. Flexagon.net probably has some rhombic patterns in there somewhere as well.
posted by soelo at 7:04 PM on September 16, 2010

Make one out of paper (or tetrahedral dice, from your favorite gaming store) held together with tape. Fold it up and color it. Then unfold it and look to see where all the colors landed.
posted by Chocolate Pickle at 7:14 PM on September 16, 2010