Close packing

[Eppur si muove]

What is the densiest space-filling polyhedra lattice?My first guess would be the cubic one, but i am not sure.

[Tide]

I think it would be hexagonal close packed (tetrahedral).

[Eppur si muove]

Can one prove that it is tetrahedral and possibly find the dimensions that would give the maximum density?

[Tide]

Yes, you can calculate the density for the various lattices and compare them. It's a fun geometry problem!

[trancefishy]

i believe it's called "cubic closest" packing, that is the best around. there is no proof that that is in fact, the best arrangement. this is an open problem. i forget what the effiiciency is, let's say, arbitrarily, it's 74% (i think that's close). it has been proven that (again, this is completely arbitrary, made up by me from loose memory) 80% is the best one can do. then again, something like 79.4% efficiency. the upper bounds are closing in, but as of yet, no proof.

EDIT: I"m a moron. nowhere did you ask about spheres. i replied about spheres. I will leave this anyways.

[Gokul43201]

The cubic close packing (or face centered cubic lattice) is the same as the hexagonal close packed (abab) structure...and is the densest 3D lattice structure. To see that the FCC is the same as the HCP, simply look at a set of parallel {111} planes in the FCC.

Trancefishy : You're not a moron. When the structure of a lattice "point" is unmentioned, it's more than reasonable to assume it is spherical.

Additional Note : There are two kinds of symmetric hexagonal close packed structures : "abab" and "abcabc". Both are equally dense, but only the abab structure is the same as the FCC.