Moulding flexural waves in elastic plates lying atop a Faqir's bed of nails

Platonic crystals (PCs) are the elastic plate analogue of the photonic crystals widely used in optics, and are thin structured elastic plates along which flexural waves cannot propagate within certain stop band frequency intervals. The practical importance of PCs is twofold: these can be used either in the design of microstructured acoustic metamaterials or as an approximate model for surface elastic waves propagating in meter scale seismic metamaterials. Here, we make use of the band spectrum of PCs created with very small clamped holes, the nails of the title, to achieve surface wave reflectors at very large wavelengths, a flat lens, an endoscope, a directive antenna near stop band frequencies and cloaking from Dirac cones. The point pinned, Faqir, plate is particularly appealing as there is an exact dispersion relation available so the origin of these phenomena can be explained and interpreted using Fourier series and high frequency homogenization.