Notably, all of these quasicrystalline phases have motifs of, or coexist with, domains of periodic tilings of squares and triangles, whose existence was hypothesized by Frank and Kasper 15, and whose topologies were first introduced by Kepler in Harmonices Mundi in 1619 16. Dodecagonal quasicrystallinity has been observed in a plethora of materials including alloys 10, 11, mesoporous silica 12, nanoparticle superlattices 13, and copolymers 14. Dodecagonal symmetry is achieved in a maximally random assembly of squares and triangles linked through four-, five-, and six-vertex nodes 8, 9. 6 or the 12-fold symmetry in dodecagonal quasicrystals 7. They exhibit symmetries that are forbidden for any periodic structure, such as the 5-fold rotational symmetry first discovered in an Al–Mn alloy by Shechtman et al. Escher.Quasicrystals are ordered materials lacking translational symmetry 1, which leads to unique physical properties 2, including novel magnetic and photonic phenomena 3, 4, 5. Tessellations figure prominently throughout art and architecture from various time periods throughout history, from the intricate mosaics of Ancient Rome, to the contemporary designs of M.C. As you can probably guess, there are an infinite number of figures that form irregular tessellations!
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