Sphere Packing, 3 Layers (C)

This is an illustration of the Sphere Packing of regulare sphere in a 3D Euclidean space.

There is two similar (as efficient) optimal ways to pack spheres in this context.
In both cases a sphere 'kisses' 12 other spheres.

In both cases, we find layers with spheres in an hegagonal pattern (A sphere 'kisses' 6 other spheres' on the same plane. All these hexagonal planes are parralel and a sphere touches 3 other spheres in the upper plane and 3 in the lower plane.

The two configurations are created whether the upper and lower plane are superposable or not.
A nice way to visualize that difference is by looking at the repartition of the 'kissing" points on the sphere.
It it the same difference as for two very similar solids : the Cuboctahedron (a Archimedean solid) and the Triangular orthobicupol (a Johnson Solid) which can be seen as a cuboctahedron twisted around his hexagonal plane.
https://en.wikipedia.org/wiki/Triangular_orthobicupola
https://en.wikipedia.org/wiki/Cuboctahedron

If the upper and lower are superposable, we have 2 layers structure where kissing points are like the Johnson solid. In this case, the only cell that can be repeated is an hexagonal prism cell.

If they are not superposable, we have a 3 layers structure where kissing points are like the Archimedean solid.
In this case, a second cell can be show. This cubic cell (square prism) corresponds to the square face of the cuboctahedron.

The 2 layers version is showing a (large) hexagonal cell with the middle (full) sphere touching 12 partial spheres. A second full sphere is shown 2 layers up with the triangular orthobicupola.

The 3 layers version is showing both hexagonal and cubic cells. On top of the large hexagonal call, a second (small) hexagonal cell is shown together with 2 cubic cells (small and large).The planes with square patterns are not parrallel to the hexagonal planes.