What are Vornoi cells for FCC and HCP Lattices ? (See plot below for FCC and HCP).
Is my uderstanding correct that we should get one of the 5 "parallelohedra" ?
If so, that seems to contradict Wikipedia: "If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: the rhombic dodecahedral honeycomb for fcc, and the trapezo-rhombic dodecahedral honeycomb for hcp." Because trapezo-rhombic dodecahedral is not the list of parallelohedra. Or I misreading something ?
FCC and HCP are densiest sphere packing lattices in R^3, which can be seen, as the same for the first two layers, but the third layer is positioned differently:
https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres
PS Motivation - to understand 3-d analogue of : https://mathoverflow.net/questions/362135/does-the-plane-clustered-to-minimize-sum-distances2-to-clusters-centers-inert