  The experimental discovery of quasicrystals by D Shechtman, D Gratias, I
Blech, and J W Cahn in 1984 provided the paradigm for a new type of long-range
order of solid matter in nature. This discovery stimulated an explosion of new
experimental and theoretical research. In years prior to the discovery, there
was a very active development of various gateways to quasicrystals in
theoretical and mathematical physics. Without this conceptual basis, it would
have been impossible to grasp and explore efficiently the structure and
physical properties of quasicrystrals. The aim in what follows is to give a
non-technical and condensed account of the conceptual gateways to quasicrystals
prior to their discovery.
