Alex's Adventures in Numberland - Alex Bellos [40]
I said that a periodic tessellation is one that repeats endlessly. There is a more practical definition of periodicity. Imagine a plane extending infinitely in all directions and covered with the triangle tessellation on the previous page. Now imagine making an identical copy of the tessellation on tracing paper and placing it on the plane. Periodicity can be defined as the capacity to lift up the copy, move it along to another position and then to put it back down on the plane so that the pattern of the copy lines up perfectly with the original pattern. We can do this with the triangle tessellation because we can move the copy to the left (or right, or up, or down) by any number of triangles. When the copy is aligned to its new position, the copy is a perfect fit for the tessellation underneath. This definition of periodicity is helpful because it is now easier to explain the concept of nonperiodicity. A nonperiodic tessellation is one that when a copy is made, there is only one position where the copy fits perfectly over the plane – the original position. For example, the tessellation below opposite is nonperiodic. (Imagine that the tessellation goes on for ever, in widening concentric pentagons.) If you made a copy of it, the copy coincides only with the underlying tessellation in one position.
Nonperiodic tessellation.
Many types of tile that can be arranged periodically can also be arranged nonperiodically. The question that tantalized mathematicians in the second half of the twentieth century, however, was whether or not there existed any sets of tiles that could be tiled only nonperiodically. These would be tiles that could cover a plane surface but were incapable of producing repeated patterns. The idea is counter-intuitive – if tiles are so well suited and harmonious that they can tile a plane without leaving any gaps, then it would seem natural that they are able to do so in a regular, repeating way. For a long time it was believed that nonperiodic tiles did not exist.
Then along came Roger Penrose with his kites and darts. In the 1970s, Penrose – a cosmologist – thrilled the maths world when he developed several types of nonperiodic tiles. The simplest were created by judiciously cutting a rhombus in two, to form two different shapes, which he called a kite and a dart. Since any four-sided shape can produce a periodic tessellation, Penrose then had to formulate a rule for how these tiles could be joined that would restrict the patterns they could make to being nonperiodic. He did this by drawing two arcs on each kite and dart and stipulating that tiles must be connected so that like arc always joins with like.
Penrose’s dart and kite can tile only nonperiodically.
The discovery of nonperiodic tiling was an exciting breakthrough for maths, but not as exciting as it would later be for physics and chemistry. In the 1980s researchers were amazed to discover a type of crystal that they did not believe existed. The tiny structure displayed a nonperiodic pattern, behaving in three dimensions just like Penrose’s tiles did in two. The existence of these structures – called quasicrystals – changed the way scientists understood the nature of matter, since it contradicted classical theory that all crystals must have symmetrical lattices derived from the Platonic solids. Penrose may have invented his tiles for fun, but they were unduly prophetic about the natural world.
Half a millennium ago, Islamic geometers might also have understood about non-periodic tessellations. In 2007 Peter J. Lu from Harvard University and Paul J. Steinhardt from Princeton claimed that their studies of mosaics in Uzbekistan, Afghanistan, Iran, Iraq and Turkey showed that the craftsmen had made ‘nearly perfect quasi-crystalline Penrose patterns, five centuries before discovery in the West’. It is possible, therefore, that Islamic mathematics may have been even more advanced than historians of science have traditionally thought.
Hinduism also used geometry to illustrate the divine. Mandalas are symbolic representations