What is a quasicrystal?

My post on 01.09.2014 about quasicrystals didn't answer the question of what exactly they are. I thought I'd remedy that now.

The atoms of most metals are arranged in a regular array, often called a crystal array (or a crystalline array), that is periodic. This simply means that if you move the array a certain distance in a certain direction, it will look the same. This is called "translational symmetry." In addition, if you rotate the array by a certain angle, it will also look the same. This is called "rotational symmetry."

This is easiest to think about in only two dimensions. That is, you can completely cover a flat space (i.e., a plane) with identical, equilateral triangles.

This called a "tiling" or a "tesselation" (not to be confused with "tesseract," a four-dimensional hypercube, or the tesseract from A Wrinkle in Time). A triangle is one of three shapes than can completely tile the plane. The other two are squares and hexagons.

(The connection with crystals comes when you imagine placing an atom at each vertex. Most crystals are in some form of cubic symmetry, but a few have hexagonal symmetry.)

In addition to translational symmetry, the triangular tiling has three-fold rotational symmetry. That is, if you rotate the triangular pattern by 120, 240, or 360 degrees, you obtain the same pattern. And, as you might expect, a pattern of squares has four-fold rotational symmetry, and the hexagons has six-fold rotational symmetry. They all also exhibit translational symmetry.

You don't have to use only one shape, you can use a repeating pattern of two (or more) different shapes.

This pattern can be moved left or right, up or down, and cover itself exactly - translational symmetry. It also has rotational symmetry. Can you see how many fold?

One thing that you CAN'T do (try it) is to completely cover the plane with pentagons, or any set of shapes with five-fold rotational symmetry. Here's an attempt at that:

If you look closely, it appears to have a five-fold rotational symmetry, but it doesn't exactly. To prove this, you'd have to print out two copies and see if you could match them up after rotation. In addition, this tiling is not periodic - that is, it is not translationally symmetric. Again, it's almost periodic, but not quite. This is the loose definition of a quasi-crystal. An array (three dimensional, of course) that has symmetry and periodicity over the short range, but not over the long range.

The picture above is of a "Penrose tiling," developed by Roger Penrose, a British physicist and mathematician. They are extremely fascinating, and I'm working on a post just about Penrose tiles. They are fun to play with, just like Zometools, and I had a lot of fun doing just that at PCMI in Utah.