Looking accurately for coordinates-based patterns

Dear all

I have created some atomic pattern of two initially superimposed layers upon twisting by an arbitrary angle. Basically, my whole system looks like

and when I zoom in the region around the middle point of the system, one can see something like

As you can see, it is possible to see some kind of patterns in the system. I would be interested in finding a square/rectangular building block defined in an orthogonal basis in the figure which I can take and replicate on the two-dimensions upon displacement along the orthogonal directions. Something like

I was thinking on something like:

(1) I take one atom in the middle of the system

(2) I take an additional atom to the right in the x-th direction. Taking this atoms I can replicate a minimal building block so that I can put it right after it and replicate what it is in the image?

(3) If no, take other atom to the left of the initial atom

Same logic along y-th direction. I would expected that it will be impossible to find an exact building block that can be replicated in space with whole accuracy, but maybe some threshold in mismatch could be implemented. Maybe some extremely small tolerance condition of 1e-6 Angstroms (which are the units of the atomic positions) deviation from orthogonal basis could work.

In the file that I am attaching, each row represent an atom of the first figure. First and second columns stand for spatial coordinates along x- and y-th coordinates, respectively, while the third column corresponds to the spatial coordinate along z-th axis. There only exist four different values along z-th. The smallest corresponds to "Atom 1", and the highest to "Atom 4".

Any ideas?

Answer 1

[Image Analyst]

This seem similar to crystallography. What I'd do is search for "x ray diffraction crystallography pattern analysis" and see what papers pop up. I remember seeing a paper at an Electron Society of America conference in Milwaukee in 1988 from David Bright at NIST that could extract multiple periodic 2-D pattern from an x-ray of a crystal - basically a complicated pattern of spots (that you could get the coordinates of). You might also explore his image analysis package Lispix:

https://www.nist.gov/services-resources/software/lispix