Alan Mackay’s contributions to non-Euclidean crystallography have furthered our understanding of quasicrystalline and icosahedral phases. He predicted the occurrence of five-fold symmetry and computed the details of the diffraction patterns to be expected from such structures several years before they were observed experimentally. He is a leading authority on the geometry and symmetry of crystals and has made pioneering contributions to periodic minimal surfaces, packing in curved manifolds, and to incommensurate structures. He has elaborated the importance of the Penrose pattern for crystallography and demonstrated optically its Fourier transform. Alan has also made noteworthy use of quaternions in crystallographic calculations.