(No, no…not the smoker’s toothpaste)

I’m having trouble envisioning something in my head, and was wondering if anyone could help.

If you take a set of circles of equal radius that do not overlap, and group them as closely as possible to each other, you get an arrangement with one in the center and six surrounding it with all borders touching. If you then “squish” them together along the axes defined by their points of contact so that the curved borders become flat faces, you get tessellating hexagons (like a honeycomb).

With me so far?

Okay. My question is: what happens when you use spheres? Spheres (on a single level) group like circles (for obvious reasons), but when you stack the layers, they are offset from each other, with each layer sitting in the “valleys” produced by the previous one.

If you again “squish” them together along all axes defined by their points of contact, what shape do they become? It seems that they should become regular polyhaedra, but there isn’t an obviously hexagonal one of those among the five Pythagoreans. Since those five are the only regular ones there are (so we’re told…hmmm :dubious: ), what am I missing?

Thanks and all.