Sure! This is highly relevant to the atomic packing factor (APF) in crystallography, which can be simplified as "If we fill a space with identical rigid spheres in highly ordered a structure, what is the fraction of volume that is occupied the spheres". To calculate APF, we usually use a unit cell to represent the space to be filled, which is the smallest portion of the space that can make up it by being repeated over and over in three dimensions. Note that spheres in a unit cell can be shared by multiple adjacent unit cells. For example, a sphere on the face of a unit cell is shared by two adjacent cells having this same face, and the space it occupies in one unit cell should be half of its volume. Here are the APFs, or packing efficiencies of some common structures:

- Simple cubic. It is a cubic unit cell with 8 adjacent spheres on its 8 vertices, and each sphere is shared among 8 cells having this same vertex. So if the radius of the sphere is r, its side length is 2r, leading to an APF of about 0.52.

- Face centered cubic. It is a cubic unit cell with 8 spheres on its 8 vertices, as well as 6 spheres on its 6 faces. You can also think of this as placing a layer of spheres on the cross points of a square grid with a side length of 2r, and placing the spheres of the second layer in the center points of the grid of the first layer, then repeating this process to fill up a space. Itis has an APF of about 0.74.

- Body-centered cubic.  It is a cubic unit cell with 8 spheres on its 8 vertices and one in its center. It has an APF of about 0.68.

- Hexagonal close-packed. The unit cell is a hexagonal prism. There are 12 spheres on its vertices and 2 on the centers of its top and bottom faces. There are another 3 sphere inside it. If we vertically divide it into 6 identical triangular prisms, these 3 spheres are in the center of 3 of the 6 prisms separated by one. You can also think of this as placing a layer of spheres on the cross points of a triangle grid with a side length of 2r, and placing the spheres of the second layer in the center points of the grid of the first layer, then repeating this process to fill up a space. It has an APF of about 0.74.

- Diamond cubic. It is also a cubic unit cell, but the locations of the spheres is hard to describe with words. An easier way is to explain the basic structure as a tetrahedron with 4 spheres on its 4 vertices and one in the center, and this structure spreads by using the 3 bottom vertices of one tetrahedron as the top vertices of another 3. It has an APF of about 0.34.
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