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How are the symmetries encoded in quasicrystals? As a compensation for thelack of translational symmetry, quasicrystals admit non-crystallographicsymmetries such as 5- and 8-fold rotations in two-dimensional space. It isoriginated from the extended crystallography group of high-dimensional latticesand projection onto the physical space having a finite perpendicular window.One intriguing question is : How does the translation operation inhigh-dimension affect to the quasicrystals as a consequence of the projection?Here, we answer to this question and prove that a simple translation infour-dimensional hypercubic lattice completely determines a distinct rotationalsymmetry of two-dimensional quasicrystals. In details, by classifyingtranslations in four-dimensional hypercubic lattice, new types of quasicrystalswith 2,4 and 8-fold rotational symmetries are discussed. It gives us animportant insight that a translation in high-dimensional lattice is intertwinedwith a rotational symmetry of the quasicrystals. In addition, regarding to themulti-frequency driven systems, it provides a unique way to discover newquasicrystals projected from the high-dimensional Floquet lattice.

Discovery of new quasicrystals from translation of hypercubic lattice