# What does the extension theorem for tilings state?

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".

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!