Hidden translational symmetry in square–triangle-tiled dodecagonal quasicrystal

We show that a two-dimensional 12-fold quasicrystal tiled with squares and triangles can be generated as a triangular periodic lattice in which the unit cell is replaced by a cluster of 19 elements defining an elementary supercell with a dodecagonal boundary. As a straightforward consequence, we obtain analytically the exact Fourier spectrum of the deodecagonal quasicrystal that can find interesting applications for modeling purposes. In perspective, since our spatially periodic assembling allows restoring Bloch-type periodic boundary conditions, photonic band gap calculations will be possible without approximating the quasicrystal geometry. We foresee extending the same basic idea to other quasiperiodic patterns