How many types of 3D packing are possible?

In my book it's said that there are 3 types of unit cubic cell - primitive , face and body centered . But for 3d packing types they have only mentioned FCC and HCP . I know that these 2 concepts- unit cell and packing types are different but it's the packing that determines the type of unit cell , right ? So are there any packing types like primitive packing and body centered packing ? .

• FCC and HCP are the two types that give 3d close packing i.e. packing atoms as densely as possible. There are many more types of unit cell, it's just that they don't achieve close packing. FCC has face-centred cubic unit cell and HCP has hexagonal unit cell. Mar 27, 2021 at 10:49
• chemistry.stackexchange.com/questions/35273/… May prove useful. Mar 27, 2021 at 15:17
• Are there any packing types like primitive packing and body centred packing ? Mar 28, 2021 at 13:59

The fcc structure is the cubic close-packed structure with highest packing density possible in 3 dimensions: $$\rho = \frac{\pi}{\sqrt{18}} \approx 74\%$$ This was conjectured by Kepler in 1611 and became a theorem during the last years by the hand of Hales (see more details here). Apart from the fcc lattice, there is another close-packed structure with the same packing density, but different arrangement of the lattice points: the hexagonal close-packing. In both types of packings, each sphere has 12 nearest neighbors and they can be understood as different two-dimensional hexagonal close-packed layers stacking in the form $$(X_{1}X_{2}X_{3}\dots)$$.
In the above figure, you can see the hcp structure at the left, with a stacking sequence $$(AB)_{\infty}$$, which is repeated infinite times over space. At the right, we have the fcc close packing with $$(ABC)_{\infty}$$, where $$A,B,C$$ means the different ways to put the hexagonal layers on top of the other ones. Well, because we have three different ways to put the layers, there are infinitely many ways to arrange the layers as the number of them is increased. All these packings are usually known as Barlow packings and they have maximum packing density, $$\sim74\%$$.
Although there are infinite of them, Barlow packings are really rare in nature. If you see the crystal structures in the Periodic Table, most of the elements are either fcc or hcp, but also bcc (a non-close-packed structure). However, some elements like Sm adopt a Barlow structure known as 9R, with a stacking sequence $$(ABCBCACAB)_{\infty}$$.