Quasicrystals are interesting quirks of solid-state physics. It used to be thought that the atoms in solid materials¹ were either totally ordered (crystalline) or totally disordered (glasses), but those in quasicrystals are neither. Unlike normal crystals, quasicrystals show order only on a small scale, but, by the same token, the order that does exist can produce large-scale effects. Quasicrystals have also, bizarrely, impacted upon the personal hygiene of many British people, once playing a key role in a vicious legal spat over toilet paper.

Ordinary Crystals

Most of us like to think we know what a crystal is. Crystals are regular arrangements of atoms arranged in such a way that symmetry is preserved². Atoms are arranged in planes and these planes are also arranged regularly with regard to each other. There are two kinds of symmetry in a crystal: point symmetry and translational symmetry. Point symmetry applies when you look at the arrangement of atoms about a point. Imagine you have the four points of a square. Now, you can rotate these through 90, 180 and 270 degrees and still end up with a shape that looks identical to the first square. Molecules often show point symmetry.

To see how translational symmetry works, imagine that you have lots of squares and you tile them to make a grid. You can move the pattern they form one or more whole squares in any of eight directions and still end up with the original pattern. Each square forms a 'unit cell', the smallest element that needs to be repeated to give the overall pattern. Crystals show this kind of symmetry on large scales as well. As the crystal grows and atoms drop into place in a regular fashion the overall shape of the crystal resembles that of its unit cell, just on a much bigger scale and in three dimensions. Salt (sodium chloride) forms cubic crystals because each unit cell is cubic. For the same reason, Alum forms octahedral crystals.

Three and Four's Company, Five's a Crowd

It's not just squares that show this behaviour, but equilateral triangles and hexagons as well. Where it begins to break down is when one tries to use pentagons. There is no way to fill a two-dimensional plane with pentagons so that no overlaps or voids occur between the shapes. So, for a long time it was believed that there were no crystals that would show five-fold symmetry for this very reason.

This viewpoint changed radically in the 1980s. One of the ways of determining what kind of symmetry a very small - say microscopic - crystal has is to put it in a vacuum chamber and bounce electrons off it. Electrons behave like waves as well as like particles, and as they bounce off a crystal's planes of atoms, they form a diffraction pattern that has the same symmetry as the crystal. It's possible to work out the exact arrangement of atoms from the diffraction pattern alone. This technique had been used for many years to do just that, but in 1984 some scientists made an aluminium-manganese alloy and found that it showed a tenfold-diffraction pattern. Theoretically, this should have been impossible, but when theory is in conflict with observations, then it's the theory that always has to give.

Theories and Tiles

Roger Penrose is a mathematician famous for, among other things, working with Stephen Hawking on the physics of black holes. He also investigated ways in which a surface might be covered by five-sided shapes. Eventually, he invented Penrose Tiling. This tiling uses two types of regular shape, rather than one, to cover a surface in a pattern that has limited five-fold symmetry.

This is how it works in its simplest case. Make two sets of rhombuses, both types of rhombus having the same length sides. One set has internal angles of 144 and 36 degrees, the other of 108 and 72 degrees. Now place them together on a flat surface so that no two shapes join together to form a regular parallelogram. You won't be able to produce any repeating patterns on a large scale but you will nevertheless be able to fill the space up. One of the things you will notice in so doing is that some of the shapes will group to form decagons, and all of these decagons will have corners that point in identical directions. You'll also notice that instead of having regular planes of atoms, as in a normal crystal, there are sets of jagged lines, but these all run more or less parallel to each other. You have just made a two-dimensional quasicrystal.

Three-dimensional quasicrystals are made in the laboratory with atoms (mainly alloys of aluminium and other kinds of atom). Icosahedra, dodecahedra and other shapes stack together to give the quasicrystalline lattice structure. Three-dimensional quasicrystals can often be sliced to reveal a two-dimensional Penrose Tiling pattern. Dan Schectman discovered the first quasicrystal in 1982, and many others have been discovered since. Single large quasicrystals have even been grown, and these show fivefold symmetry.

Quasicrystals tend to be quite hard substances, but they behave like metals in many ways. They are, however, not very good at conducting heat and electricity. This is hardly surprising given the absence of any long-range order: metals conduct heat and electricity through their free electrons, and in quasicrystals these are scattered very frequently.

One of their most interesting properties arises directly from their aperiodic nature. Friction is thought to arise from direct contact between two periodic atomic arrangements, like two saw blades rubbing over each other. Quasicrystals are not periodic, so they don't stick to anything, rather like Teflon. To make a quasicrystalline frying pan, you electrodeposit tiny particles of Al₆₅Cu₂₃Fe₁₂ using dissolved nickel on the pan's surface and standard electroplating equipment. You then polish and roughen the surface to produce a frying pan that won't wear out and won't poison you if you leave it overheating.

Lowering the Tone

And the toilet paper? Roger Penrose was not pleased at all when his wife brought home a pack of two-ply toilet rolls decorated with the two-rhombus tiling pattern. In fact, he regarded this as the scientific equivalent of daubing creosote over the Mona Lisa, and filed a lawsuit against the manufacturer for breach of copyright. The manufacturer pleaded that the aperiodic pattern stopped the layers of paper from sticking together³. Eventually Sir Roger decided not to 'sue the ass' off the company, instead coming to a mutually beneficial arrangement. If the company had thought hard, however, they could have pleaded that their pattern was not truly a 'rip-off' of Penrose Tiling, as it would have had to have shown some periodicity in order to be mass-produced. A little learning is indeed a dangerous thing, especially when bottoms and mathematics come into contact.