No, your question was stated reasonably well; Rachm Qoch was able to understand you. I just didn’t read it carefully enough.
What I think you were asking was for the set of all points closest to a given sphere in the packing. (Think about the two-dimensional case; but instead of “squishing the circles together” imagine them inflating until they meet a neighbor and then flattening out. When all the plane is filled, each region will consist of the set of points closest to a given center. These are the hexagons you’re talking about.) This region is useful in the study of lattices, so it has a name; it’s called the “Voronoi cell” for the arrangement of points (or spheres, in this case).
I’ll try to draw a picture to show the different packing arrangements:
A A A A
B C B C B
A A A A A
C B C B C
A A A A
B C B C B
A A A A A
All of the "A"s are arranged in a planar hexagonal lattice. The "B"s show one possible position for the second layer, lying in (half of) the dimples on top of the A lattice; the "C"s show the other possible position (using the other half of the dimples). I’ve left some of the positions unlabeled so you can see the A lattice better. The B and C lattices are also hexagonal lattices; they’re just shifted in position. So if the first two layers are an A layer and a B layer, the third layer can then be either A (directly above the first layer) or C (not directly above the first layer).
Again, it depends on what angle you look at it. The rhombic dodecahedron does have some hexagonal cross-sections (it also has some square cross sections). You know how, if you look at a cube directly at one corner, the cube looks like a hexagon? Well, if you look at a rhombic dodecahedron from that same vantage point, you’ll also see a hexagon. Those are, in fact, the hexagons that correspond to the 2-d hexagonal tiling, just as you expected.
This would be so much easier if you were standing at my desk and I were trying to explain in person… As it happens, I do actually have a rhombic dodecahedron on my desk. It’s a lot easier to understand this if you have one you can pick up and hold and turn in your hands. Mathworld’s entry includes a Java applet that you can rotate freely; it might help a bit.
Ah…that starts to make sense (although at first glance, I didn’t “want” a dodecahedron, I just wanted whatever resulted from this exercise). If I was undully sniping at you, you have my apologies. You were rather bewilderingly harsh in another math-oriented GQ thread of mine, and I assumed you were doing the same thing here.
Imagining them inflating works just as well for me. It seems to produce the same result. I’m off to Google “Voronoi cell.” This kind of stuff is fun; I wish it made more sense to me.
Hey, I’d welcome the opportunity, because I think you’re right. Do you remember where you picked up the model? I would love to have one.
No, actually it was about set theory. I’m trying to find it, but the hamsters are on coffee break or something. No big deal in the grand scheme of things, but it sure irritated me at the time.
Yes, but it was at a garage sale, so that probably doesn’t help you much. It’s a sort of toy thing made of pieces hinged together with plastic-paper: You can fold it into a cube, or turn it inside-out into a rhombic dodecahedron, or a bunch of other shapes. One of the faces is labelled “Yoshimoto”, but I’m not sure if that’s the name of the specific toy, or the company that makes it, or what.
I’ve been looking for a reason to reply to this thread since it caught my eye. As (a) topologist, I actually don’t have anything interesting to add beyond what’s been said. However: Is this what you have? There seem to be quite a few references on the web to “Yoshimoto cubes”. Yoshimoto appears to be the designer who came up with this.
No, that’s not what I have, though a friend had one of those when I was a kid, and there’s a definite kinship between them. My thingie is all one piece (though it’d work interestingly with another identical one), and is less starry-shaped. In cube form, it’s solid, and in dodeca-form, it’s got a hollow cavity in the middle the size of the cube. Further searching brings up a description, listed on page 13 of that document, what he calls a “kaleidoform”. But I haven’t been able to follow that up any further.
