DanA's great-circle/Voronoi argument seems to give 4(n choose 2) as the number of faces created by intersecting n cylinders.
n 2 3 4 6 10 F 4 12 24 40 60 84 112 144 180
A 180-hedron! Is there any limit to the number of congruent faces on a polyhedron? E.g., recursively erecting pyramids of the right shape gives the sequence tetrahedron, cube, rhombic dodecahedron, then our recent friend, the deltoidal icositetrahedron. But the succeeding 48-hedron has faces of two types.
Perhaps Eric's Rhombic Dodecahedron article http://mathworld.wolfram.com/RhombicDodecahedron.html could mention that if you place a nitrogen at one of the eight vertices of valence three (which form a cube) and hydrogens at the adjacent vertices, you get an anatomically correct ammonia molecule. I.e., those four vertices form the center and three vertices of a regular tetrahedron.
Tetrahedral minifact: Resting on a flat surface, the (dihedral) angle between the surface and a (sloping) face equals the bond angle (between rays from the center to two vertices), namely pi - asec 3. Is this equality geometrically obvious? --rwg REPLICATILE PERCIATELLI