So what's the big deal? A quasi-crystal is to a crystal, what a transcendental number is to a repeating decimal. A fraction like 1/3 written out never ends, but it never changes, just like ordinary squares cover a flat surface, by repeating over and over again, or cubes fill a space. Quasi crystals fill space completely, but do not repeat, even though they show self-similar patterns, the way pi has order, but doesn't repeat. That is, the tessellate in an ordered way, but do not have repeating cells.
In art Girih tiles showed the essential property of being able to cover an infinite space, without repeating. In mathematics, Hao Wang came up with a set of tiles that any Turing Machine could be represented by, and conjectured that they would eventually always repeat. He turned out to be wrong, and over the next decades, tiles that did not repeat, but showed order, were discovered, most famously, though not first, by Penrose.
Physically, when x-rays diffract, that is are scattered, from a crystal, they form a discrete lattice. Quasi-crystals also have an ordered diffraction pattern, and it tiles the way ordered by non repeating tiles do. Quasicrystal patterns were known before Schechtman labelled them.
So why care? Because crystals have only certain symmetries, and that determines their physical properties. Quasicrystals can have different symmetries, and do not bind regularly, and so different physical properties – which means new kinds of materials. Some examples are for highly ductile steel, and in something that is a bit of a by-word among people who study them cooking utensils."
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