# Periodic boundary condition - 26 translated images

Hi all,

Could someone please help explain why each rhombic dodecahedron simulation box (unit cell) is surrounded by 26 translated images?

I’ve tried, but still failed to visualize the arrangement of these periodic boxes.
Should there be 12 boxes instead, since a rhombic dodecahedron has 12 faces, and 26 sides.

Thanks a lot,
Anh T. Mai

Where have you read about it being 26?

The GROMACS manual notes it is 12, and this visualisation shows that it is 12 as well.

Hi Dr_DBW,
The animation of the visualization is great.
Yes, 12 faces, 24 edges and 14 vertices. I was wrong and miscounted the edges (the sides).

Originally, I read it from the GROMACS Documentation, Release 2019, by Dec 31, 2018 (the .pdf version of the manual).

It’s also written in the online manual page that you shared, the last paragraph,
“Each unit cell (cubic, rectangular or triclinic) is surrounded by 26 translated images. A particular image can therefore always be identified by an index pointing to one of 27 translation vectors and constructed by applying a translation with the indexed vector (see Compute forces). Restriction (5) ensures that only 26 images need to be considered.”

Yes, thanks for pointing out the statement as “Each of the 12 image cells is at the same distance.” (in the paragraph right after the box). I didn’t pay close attention to it.

By the way, the angle between ab, of the rhombic dodecahedron (xy-square), should be 90 degrees, right? I found in another version of the manual, it was noted as “90” degrees.

Many thanks,
Anh

That refers to the cubic box. Each cube is in contact with other 26 cubic boxes. Of these, 6 are in face contact, 12 in edge contact and 8 in vertex contact.
Each rhombic dodecahedron box is surrounded by 18 translated images, not 26. Of these, 12 are in face contact and 6 in vertex contact. Any translated image sharing an edge with the simulation box also shares a face with it, so there are no images in edge contact only.

Atoms in any periodic unit cell can interact with a least 3^3-1=26 periodic images, that is periodic images that are one periodic shift away along either x, y and/or z. Under some circumstances a rhombic dodecahedron can interact with a periodic image shifted 2 periodic vectors along x, so even more than 26 images.

This is a little confusing (to me at least). A rhombic dodecahedron is not a unit cell. Unit cells are always parallelepipedal.
For clarity of explanation, I will define a “neighbour” unit cell. A neighbour unit cell is an image parallelepiped of the fundamental unit cell that shares at least one of its elements with the fundamental unit cell. This definition is equivalent to saying what you write: one periodic shift away along x, y and/or z.
If the parallelepiped is so skew that the cutoff distance for pairwise interaction is larger than the shortest dimension of such a skew unit cell, then I can see how an atom could be interacting with another atom contained in a non-neighbour unit cell. If this happens, I would think that that atom would also be interacting with one of its own images (in the neighbour unit cell in their midst).
Am I talking nonsense?

A single atom will never interact with all 26 periodic image cells. But in most cases there are interactions with all 26 nearest periodic image cells with some of the atoms in the system. This can be even more for some very skewed unit cells.