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Consider two periodic lattices of points extending to infinity and the question of commensurability of the two.

If for instance we take one of them to be a square lattice called $S_1$, it is fairly easy to see that any other square lattice $S_2$ with basis vectors being integer multiples of those of $S_1$ will be commensurate with $S_1$.

Similarly, given a triangular lattice $T_1$, any other lattice $T_2$ with basis vectors being multiplied by the square root of a Loschian number is commensurate with $T_1$ (Loschian numbers being numbers that can be expressed as $a^2+ab+b^2=1,3,4,7,9,12,13,$..., with $a$ and $b$ integers).

However there exist certainly many more interesting scenario: for instance the square lattice is commensurate with the snub-square lattice, and the triangular lattice is also commensurate with the honeycomb lattice, the kagome lattice, the snub trihexagonal lattice etc. In particular these lattices are all characterised by different density of points per unit area.

Is anyone aware of a list of possible lattices commensurate with each other?

I am interested in anything concerning the problem but I would be particularly looking for a list of lattices commensurate with the triangular lattice.

semola
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I would suggest using the Smith normal form of the Gram Matrix of the lattice. Similar matrices have the same smith normal form. Additionally, if the matrices are not similar, you can check if they have many points in common, that is if they are sublattices of each other, or if they overlap in some ways, but not in others. That way it should be also possible to prove hypotheses like the one about Loschian Numbers

user222134
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