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More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $n$?

Question. For what values of $n$ can we find such tilings?

I have started to work on this problem. From the literature, I found examples for $n=5$ (Penrose tiling) and $n=7$ (Madison's $7$-fold). Then, from my own investigation, I found examples for $8$, $10$, $12$, and $16$.

Here are pictures of these tilings:

$n=5$

Penrose

$n=7$

Madison's 7-fold

$n=8$

patch_8

$n=10$

patch_10

$n=12$

patch_12

$n=16$

patch_16

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    $\begingroup$ I think one of the Penrose constructions generalizes to any n: start from the action of Z/nZ on Z^n by cyclic coordinate permutations; choose a 2-dimensional subspace V in the ambient R^n on which Z/nZ acts irreducibly; and for some d>0 let C be the set of projections to V of lattice points that are at distance at most d from V. The tiling is then the Voronoi tiling associated to C (or the dual Voronoi tiling, or other variations). $\endgroup$
    – Noam D. Elkies
    Commented Dec 5 at 2:04
  • $\begingroup$ I think subrosa does this for even $n$ link.springer.com/article/10.1007/s00454-016-9779-1 $\endgroup$
    – Ville Salo
    Commented Dec 5 at 6:57
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    $\begingroup$ Please see in "Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings" by Gähler/Rhyner Sect. 8 "Symmetric two-dimensional grids" where you can find the construction very explictly: math.uni-bielefeld.de/~gaehler/papers/jav19i2p267.pdf. (By the way, the minimal dimension for embedding a $n$-fold symmetry in a periodic lattice is given by the Euler's totient $\phi(n)$ with e.g. $\phi(5)=\phi(8)=\phi(10)=\phi(12)=4$, so quasiper. tilings of these rotat. symmetries can be obtained by cut and projection from 4-dim. lattices.) $\endgroup$
    – Andreas Rüdinger
    Commented Dec 5 at 22:01
  • $\begingroup$ @VilleSalo Please will you re-post your comment as an answer, instead of just a comment? Then I can up-vote it and mark it as correct. As I see it, this is the complete answer to the question. $\endgroup$
    – Andrew Bayly
    Commented Dec 7 at 0:16

1 Answer 1

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Jarkko Kari and Markus Rissanen construct such (even a substitutive one), called Sub Rosa, for even $n$ in [1]. ArXiv is https://arxiv.org/abs/1512.01402

The second-named author is Markus Rissanen, who is not a mathematician but an artist. He is credited as the inventor of the tilings in https://dimensiolehti.fi/laatoitusten-geometriaa/

[1] Kari, Jarkko, and Markus Rissanen. "Sub Rosa, a system of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry." Discrete & Computational Geometry 55 (2016): 972-996.

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