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The icosahedral tiling modulo the unit cell of the PAS results in point sets circumscribed by triacontahedra on a fcc lattice (a). The QPNC consisting of steel beads in polyester (b). Shapes of the (pseudo) zone boundaries of the fcc (c) and the icosahedral (d) structures. Experimental longitudinal wave transmission spectra of the QPNC along a twofold axis (solid line) and its APNC for the  direction (dashed line) (e). Calculated band structure of the APNC along the  direction (f). Transmission (dashed line) and reflection (dotted line) spectra for shear waves for the APNC (g) and QPNC (h) [directions as in (e)]. The shaded regions in (e)–(g) denote the frequency ranges of the longitudinal hybridization and the shear Bragg gaps.
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Icosahedral band gap materials (BGMs) optimally combine the distinct band gaps of periodic BGMs with the high rotational symmetry of quasiperiodic structures. This is shown experimentally for longitudinal and transverse polarized elastic waves in a phononic crystal based on the three-dimensional Penrose tiling (3D-PT) and applies equally to photonic crystals. The ability of icosahedral BGMs to form Bragg-type band gaps follows from the similarity between the 3D-PT and the face-centered cubic structure (its periodic average structure). The 3D quasiperiodic BGM lacks bands of strong transmission like random or disordered BGMs but shows clear band gaps like periodic BGMs do.
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