This invention relates generally to the field of quasicrystalline structures. In preferred embodiments, the stopgap structure is more spherically symmetric than periodic structures facilitating the formation of stopgaps in nearly all directions because of higher rotational symmetries. More particularly, the invention relates to the use of quasicrystalline structures for optical, mechanical, electrical and magnetic purposes. In some embodiments, the invention relates to manipulating, controlling, modulating and directing waves including electromagnetic, sound, spin, and surface waves, for pre-selected range of wavelengths propagating in multiple directions.