Periodic Wang tiling implies existence of doubly periodic tiling. I'm struggling to understand how to prove a theorem stated by Karel Culik in his paper on 13 aperiodic Wang tiles. He defines a periodic tiling over a finite tile set T as being a function $f:\mathbb{Z}^2 \rightarrow T$ such that $f(x,y)=f(x+h,y+v)$ for some $(h,v)\in \mathbb{Z}^2$, with $h,v \neq 0$.
He then says that if $T$ admits a periodic tiling, then there exists a doubly periodic tiling with tiles in $T$, i.e. a tiling $g$ such that $g(x+h,y)=g(x,y)=g(x,y+v)$.
Is the proof of this trvial, and I'm overthinking it? Or is there simply something I'm missing?
Thanks in advance.
 A: Lets do the case where the tiling $\mathcal{T}$ is periodic in the horizontal direction, so $\mathcal T+(k,0) = \mathcal T$ for some $k \geq 1$.
That means, in particular, that each horizontal strip in the tiling is $k$-periodic. There are only finitely many possible $k$-periodic sequences over the set of prototiles, and so there are only finitely many possible horizontal strips in $\mathcal T$. Label them $\mathcal S_1, \mathcal S_2, \ldots,\mathcal  S_m$. For any particular pair of strips $\mathcal S_i$, $\mathcal S_j$, we can check whether $\mathcal S_i$ is legally allowed to appear above $\mathcal S_j$ by just checking a length-$k$ block in each, say from the $0$th position to the $(k-1)$th. In particular, it means that we can build a graph whose vertices are the $S_i$s and where a directed edge goes from $\mathcal S_i$ to $\mathcal S_j$ if $\mathcal S_i$ can legally appear above $\mathcal S_j$.
It's not too hard to see, I hope, that the existence of a doubly periodic tiling is now equivalent to this graph admitting a cycle. But of course, the graph has only finitely many vertices, and we know that there exists at least one bi-infinite path in the graph, because that path corresponds exactly to the tiling $\mathcal T$! It follows that the graph admits a cycle and so there exists some doubly periodic tiling using the wang tiles.
The non-horizontal case is similar - one just replaces the horizontal strips by $(p,q)$-staircases, where $(p,q)$ is a known direction of periodicity.
