[kkylin]

In addition to the above links on quasicrystals etc, it may help to have a bit of context on periodic tilings, which have very precise mathematical properties: https://en.wikipedia.org/wiki/Lattice_(group).

The structure of the periodic table itself can also be understood, to leading order anyway, in symmetry terms. See for example https://mathstodon.xyz/@johncarlosbaez/112540366778806757 and references there.