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Suppose you have a quasicrystal where each layer has the wall paper group of a square tiling and is the same as the layer below it except that layer is rotated counterclockwise from the previous layer by an angle of $\sin^{-1}(3/5)$. I think that in theory can happen because $(3, 4, 5)$ is a Pythagorean triple. Then since that angle is an irrational amount of a full revolution, every layer would have a different orientation so at the macroscopic level, the properties would not vary at all with any amount of rotation about an axis perpendicular to a layer.

Is such a quasicrystal known to actually exist?

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