the object in question is a truncated icosahedon whose sides are pearls.

It is an interesting little bauble which can be found in some cheap jewelry stores.

Each pearl is drilled and through the hole thus created there are two threads.

Upon exiting the hole (therefore at a vertex) the threads are directed towards adjacent pearls.

Since at each vertex three sides (pearls) come together, there are three threads at at each vertex (these threads are visible if one looks at the picture carefully).

Any idea how this "threaded" truncated icosahedron was put together?

I suspect this may have to do with graph theory, an assumption very easy to make since I know absolutely nothing about the subject.

My ultimate goal is to be able to put together such a polyhedron using small pieces of pipes and string, and who knows may be other polyhedras.

I have built a truncated icosahedron with Zometool parts but I have not been able to find a solution, only conjectures.

Thank you.

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I see an uncommented vote to close, which is inexplicable to me. I would guess that a substantive (versus coincidental) answer might involve both representation theory and graph theory. The question is sufficiently clear to understand. I hope any other votes to close have a good and articulated reason. – Steve Huntsman Jun 4 '10 at 19:53
Besides being cubic, the truncated icosahedral graph is Hamiltonian: mathworld.wolfram.com/TruncatedIcosahedralGraph.html – Steve Huntsman Jun 4 '10 at 19:55
I'm starting to get it. I've been staring at the picture, if we regard the pearls as vertices the figure is what you get if you take the truncated icosahedron and truncate again at each vertex as far as the midpoint of each edge, making for a large number(60) of additional equilateral triangles, while still having 20 hexagons and 12 pentagons. I don't know a name for the solid I am describing...You prefer to view each pearl as being the midpoint of an edge. – Will Jagy Jun 5 '10 at 4:10