Nobel Prize in Chemistry for quasicrystals

NHPR announced the Nobel prize in chemistry was given to Daniel Schechtman of Technion in Israel for the discovery of quasicrystals.
I never heard of quasi crystals before so I did some research. Far as I can tell, quasi crystals are a form of solid matter in between regular crystals and glasses. Glass lacks any kind of order on the molecular level; glasses are more like a supercooled liquid than a crystalline solid.
Quasi crystals are somehow less ordered than ordinary crystals. When they refract X rays they don't show some of the symmetries that regular crystals do. Just what those symmetries are, and what they mean was not clear to me.
Since most of what we know about the crystal structure of matter comes from X ray diffraction experiments, discovery of a new structure is clearly very important in the field of X ray crystallography. Applications outside that field are less obvious right now, although we can expect it to lead to better insight into strength of materials, and possibly lead to stronger building materials or stronger metal alloys.