non-trivial non-repetitive aperiodic tiling of the plane Which is the less trivial example of non-repetitive aperiodic tiling of the plane you know? I cannot come up with a famous non-repetitive tiling. Are there any? A tiling is repetitive if for every pattern $P$ there's a radius $R$ such that for every point $x$, the ball $B_R(x)$ has a translate of $P$.

The most simple? I guess take any Penrose tiling (or some other repetitive aperiodic tiling) and replace a single tile with the same tile but of a new colour, or if you want tiles to not be able to be labelled, just add a notch to one edge of a tile-tile intersection. This is a rather boring answer, but probably the simplest example.

If you want some kind of stronger property - say you want that for every patch $P$ there exists **no** $R$ such that every $x\in\mathbb{R}^2$ has the patch $P$ somewhere in the ball $B_R(x)$ - then perhaps a Pinwheel tiling is what you're after. It satisfies the above stronger non-repetitivity condition with respective to translations, but it is repetitive with respect to _Euclidean motions_ (where you allow for translations follows by a small rotation).

(Credit for first image - Charles Radin)

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