What does the extension theorem for tilings state?

Question

I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space. E.g. in "The Local Theorem for Monotypic Tilings" one reads

The Extension Theorem [...] gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space.

I have a hard time tracking down the exact statement of this theorem. I found some sources (see below), but these are available only in Russian (despite the English titles).

• N. Dolbilin, "The Extension Theorem".
• N.P. Dolbilin and V.S. Makarov, "The extension theorem in the theory of regular tilings and its applications".

Answer 1

The first source is in English in Discrete Mathematics. You can find it here

• Nikolai P. Dolbilin, The Extension Theorem, Discrete Mathematics 221 Issues 1–3 (2000) pp 43–59, https://doi.org/10.1016/S0012-365X(99)00385-4.

Answer 2

I first read about the extension theorem for tilings in a simpler form: if a finite protoset can tile an arbitrarily large disk, then it can the whole plane. My informal way of proving it is the following. Let $P$ be a finite set of prototiles. Consider a sequence $(C_n)_{n\in\mathbb{N}}$ of partial coverings of the plane such that $C_n$ covers a disk of radius $n$. Observe that there must be a tile $T\in P$ appearing infinitely often in $(C_n)$, say exactly in the subsequence $(C_{n_k})$. Now, it seems reasonable to assume that finitely many prototiles can be arranged in only finitely many ways. Therefore of all possible 1-coronas around $T$, one of them, say $A_1$, must appear infinitely often in $(C_{n_k})$, say exactly in the subsubsequence $(C_{n_{k_\ell}})$. Now of all possible arrangements of prototiles that coronate $A_1$, one of them, $A_2$, has to appear infinitely many times in $(C_{n_{k_\ell}})$, say exactly in the sub³sequence $(C_{n_{k_{\ell_m}}})$. Repeating this argument infinitely many times provides the existence of an unbounded sequence of arrangements, $(A_n)_{n\in\mathbb{N}}$, each of which is a coronation of the previous one. It is obvious that $A_n$ converges to a covering of the plane!