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

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  • $\begingroup$ This was amazingly helpful. If you could also include (a simplified version of) the statement of the theorem as well, I would immediately accept this answer. Otherwise, if you don't mind, I am going to work out a version myself. $\endgroup$
    – M. Winter
    Commented May 19, 2020 at 21:04
  • $\begingroup$ In simple words, the theorem says that if we have a convex polytope P and that if for some k>0 we can 1) Surround P with k layers of congruent copies of P (so-called k-corona); 2) The symmetry group of k-corona is the same as the symmetry group of (k-1)-corona 3) For each neighbor Q of P, there is an isometry that moves P to Q and agrees on their coronas, Then we can extend these k-layers to a tiling of the whole space. And there are also properties of this tiling too. $\endgroup$
    – A. Garber
    Commented May 25, 2020 at 2:31
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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!

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  • $\begingroup$ I think this is usually called compactness rather than an extension theorem. $\endgroup$
    – Ville Salo
    Commented May 24, 2023 at 13:04
  • $\begingroup$ Thank you. It does seem so. $\endgroup$
    – aleph2
    Commented May 24, 2023 at 13:12

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