Much of chemistry is a closed book to me: I took the required one term of general chemistry in college (more or less an overview of the field), but that was it. It’s not snobbery, more that chemistry is a very experimental science, and I’m not very good in the lab. However, I was pleased to learn that this year’s Nobel Prize in Chemistry is for a discovery I know a tiny bit about, so I can hopefully convey some aspects of why it’s interesting.

Solid materials are generally crystalline in character: the atoms or molecules are spaced in a predictable way. There are many types of crystals, based on how the atoms are arranged with respect to each other. The key to classification is symmetry: if you move along a crystal in a particular direction for a specific distance, the crystal will look the same (until you hit the edge, which is another area of study). For simplicity, I’ll mostly confine the discussion to two dimensions; extension to three dimensions is slightly more complicated, but follows the same principles.

If each atom is connected to four other atoms at equal distances, you get a square lattice: moving along the lattice in the directions shown in the figure will take you to an identical section of the crystal. (If the bond lengths in the two directions aren’t equal, you’ll get a rectangular lattice instead.) If each atom in a two-dimensional lattice is connected with three other atoms at equal distances, you get a hexagonal lattice, with different directions of symmetry. Graphite, a form of carbon, is made of sheets of hexagonal lattices stacked on top of each other; the bonds within the sheets are strong, while the bonds between the sheets are weak, which means the layers can easily slide across each other. That’s why graphite is a good lubricant, and why it is a major component of pencils.

In both of these cases, an easy way to construct the total pattern is using tiles, as shown. The tiles all have an atom or molecule at their center, and the shape of the tile is the same as the name of the lattice pattern: square or hexagonal. (Note to get a triangular lattice, you need to connect each atom to six others.) In real crystals, the patterns don’t continue forever, of course, and can be complicated by fractures and distortions. The basic picture is clear, however: the atoms seek the strongest bonds, which means they tend to collect according to simple rules of symmetry, which in turn makes simple geometrical shapes.

The hexagonal lattice involved three connections; the square lattice involved four. What about five, with a pentagonal tile? It turns out you can’t make a crystal that way! Pentagons of equal sizes can’t be fit together in the same way: there will always be empty spaces in the lattice that aren’t of the same shape or size as the tiles. Islamic artists in the 15th century discovered that they could tile pentagons with 10-pointed stars, but if they did that, the pattern didn’t repeat itself. A more rigorous mathematical study was made by mathematical physicist Roger Penrose, who found a simple tiling pattern of four-sided shapes of two sizes; again, Penrose tiles don’t form repeating patterns. (Penrose and his father were friends with the great M.C. Escher, who loved working with tiling patterns in his art.)

Penrose tiles and the Isfahan mosaic can be contrasted with regular crystals like this: both have local symmetries, appearing very regular when you look at them up close. However, when you look at the big picture, zooming out, crystals appear to be the same everywhere, while any kind of long-range coherence is lost in the case of Penrose tiles. Since solids are practically defined by their crystalline structure, it was pretty evident to most scientists that the Penrose tiles were nothing but mathematical and artistic playthings, interesting, but unconnected with nature. In three dimensions, a similar situation occurs with shapes known as icosahedrons, which have 20 triangular sides: again, they have five-fold symmetry, but you can’t pack them together without leaving empty spaces.

In 1984, Daniel Shechtman and his colleagues found a non-repeating lattice in an alloy of aluminum and manganese that exhibited five-fold symmetry. They discovered this odd property using electron diffraction: electrons are fired at the material and their wave characteristics create an interference pattern revealing the positions of the atoms. D. Levine and Paul Steinhardt (whom I know mostly through his work in cosmology) showed that the diffraction pattern was consistent with packing of icosahedrons with other shapes. Since this kind of thing obviously isn’t a crystal in the usual sense, they dubbed it a quasicrystal. Since then, quasicrystals have been found with eightfold and even twelvefold symmetries, but Shechtman’s initial discovery was treated with suspicion by many of his colleagues.

Of course, we know and celebrate the end of the story: today, Daniel Shechtman has been awarded the 2011 Nobel Prize in Chemistry for his discovery of quasicrystals.

Update: while I was writing this post, Scientific American re-ran the late great Martin Gardiner’s last column on recreational mathematics, which among many other things discusses Penrose tiles and quasicrystals.