Here is a somewhat unusual kind of symmetry, that of a quasi-crystal .
To set up this discussion, you have probably seen various tilings of the plane possessing symmetries by discrete groups: \(\mathbb{Z}/2, \mathbb{Z}/3, \mathbb{Z}/4, \mathbb{Z}/6\) ( \(\mathbb{Z}/n\) refers to rotations of \(2\pi/n\) . The \(\mathbb{Z}/2\) group, therefore, relates to symmetrical rotations of \(\pi\) radians or 180 degrees. \(\mathbb{Z}/3\) , similarly, relates to rotations of 120 degrees; \(\mathbb{Z}/4\) relates to rotations of 90 degrees, and so forth). 1 See Fig. 17 .
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| \(\mathbb{Z}/3\) | \(\mathbb{Z}/4\) | \(\mathbb{Z}/6\) |
