Consider the plane tiling shown below. All polygons are regular with side $1$. The tiling is centered at the origin. In terms of complex numbers, the vertices are in $\mathbb Z[\omega,i]$, where $\omega$ is a primitive $6$-th root of unit, and so can be written uniquely as an integer linear combination of $1,\omega,i,\omega i$. However, not all such linear combinations correspond to vertices of the tiling. In fact, $\mathbb Z[\omega,i]$ is dense in the complex plane.
Do the symmetries of the tiling induce an algebraic structure on the set of vertices?
Can we characterize the set of vertices inside $\mathbb Z[\omega,i]$ ?
Here is some data to get you started:
The central hexagon has vertices $1, \omega, -1+\omega, -1, -\omega, 1-\omega$.
The square vertically adjacent to the central hexagon has vertices $\omega, \omega+i, -1+\omega+i,-1+\omega$.
Multiply these repeatedly by $\omega$ to find the vertices of the other squares adjacent to the central hexagon. Use that $\omega^2=-1+\omega$.
All vertices in the tiling are translations of these central vertices by integer multiples of $-1+2\omega+2i$ and $1+\omega+2i-2\omega i$.
Thus, a not very satisfactory answer to my first question is that the set of vertices is a union of cosets. But I'd like a nice answer to the second question in terms of exactly which integer linear combination of $1,\omega,i,\omega i$ occur.
My answer to the first question remains: the set of vertices is a union of cosets, as illustrated below (the coset representatives are shown in blue).
This is the basis of an integer representation for periodic tilings of the plane by regular polygons, as explained in a paper that has just appeared online. A preprint is available here. Explore a large collection of tilings here.
The vertices in all such tilings can be given integer coordinates in $\mathbb Z[\omega]$, where $\omega$ is the principal 12th root of unity. These coordinates form a $(2+n)\times4$ integer matrix, where $n$ is the number of cosets in the tiling and the $2$ comes from representing two translation vectors that define a translation cell for the tiling (the red parallelogram in the picture). The cosets are with respect to the discrete subgroup generated by the translation vectors.
The question of which integer matrices are possible appears to be delicate and is related to the enumeration of all possible tilings. Essentially, the area of the parallelogram has to be an integer combination of the area of a square and of the regular triangle.