[smallnamespace]

For crystallography, the use of group theory in part originates in X-ray crystallography [1], where the goal is to take 2D projections of a repeating 3D structure (crystal), and use that along with other rules that you know to re-infer what the 3D structure is.

Repeating structures have symmetries, so seeing the symmetries in your diffraction pattern inform you of the possible symmetries (and hence possible arrangements) in your crystal. Group theory is the study of symmetry.

By the way, this is also how the structure of DNA was inferred [2], although not from a crystal.

[1] https://en.wikipedia.org/wiki/X-ray_crystallography#Crystal_...

[2] https://www.dnalc.org/view/15014-Franklin-s-X-ray-diffractio...

[adw]

> use that along with other rules that you know to re-infer what the 3D structure is

Great answer, thank you :-) Saved me a bunch of typing to explain it less well than you just did.

It's worth adding, for this crowd, that another way of thinking about the "other rules" you allude to is as a system of constraints; you can then set this up as an optimization problem (find the set of atomic positions minimizing reconstruction error under the set of symmetry constraints implied by the space group – so that means that solving crystal structures and machine learning are functionally isomorphic problems.

[soVeryTired]

I thought the work on the structure of DNA used Fourier analysis more than group theory.

I know harmonic analysis in general combines the two, but I'm sure Crick and Watson could have done their work without knowing the definition of a group.

[adw]

And by Crick and Watson you mean Crick, Watson, Franklin and Wilkins, right? It's fairly clear all four deserve at least partial authorship by modern standards. James Watson was a piece of work.

(https://www.theguardian.com/science/2015/jun/23/sexism-in-sc...)

Crick was absolutely certainly familiar with the crystallographic space groups; he was the student of Lawrence Bragg (https://en.wikipedia.org/wiki/Lawrence_Bragg), who is the youngest ever Nobel laureate in physics – winning it with his father for more or less inventing X-ray crystallography. It's mostly 19th-century mathematics, after all.