Quasicrystals – the forbidden five
Crystals cannot have fivefold symmetry, crystallography taught. In 1982 Dan Shechtman found it nonetheless. The order behind it is that of the Penrose tiling and the golden ratio.
Pentagons cannot tile the plane without gaps in a regular pattern. That is one of the oldest insights of geometry, and on this site it is the mark of the pentagram as the “boundary crosser” (see Circle, triangle and square). Crystallography made a law of it, the crystallographic restriction: a crystal whose pattern repeats regularly can only have symmetries of order 1, 2, 3, 4 and 6. Five is forbidden.
The discovery
On 8 April 1982 the Israeli materials scientist Dan Shechtman examined a rapidly cooled alloy of aluminium and manganese under the electron microscope. The diffraction pattern showed ten bright spots in a circle, a tenfold and thus fivefold symmetry. In his notebook he wrote “10 fold???”. According to accepted doctrine, no such thing could exist.
The scientific world was dismissive. Shechtman was asked to leave his research group, and the two-time Nobel laureate Linus Pauling scoffed that there are no quasicrystals, only quasi-scientists. Only in 1984 could Shechtman publish with colleagues. Gradually other laboratories found the same, in 1992 the International Union of Crystallography changed its definition of a crystal, and in 2011 Shechtman received the Nobel Prize in Chemistry. In 2009 a natural quasicrystal was even found, in a meteorite from the far north-east of Russia.
Order without repetition
The solution to the puzzle: quasicrystals are strictly ordered, but their pattern never repeats exactly. The mathematical model had already been found by the physicist Roger Penrose in 1974. With just two rhombi, a thick one with angles of 72° and 108° and a thin one with 36° and 144°, the plane can be tiled without gaps, but never periodically. Everywhere one finds fivefold stars and decagons.
A Penrose tiling of thick (gold) and thin (blue) rhombi. Each step divides the rhombi by fixed rules into smaller ones. The ratio of thick to thin rhombi approaches the golden ratio.
The golden ratio is everywhere in this tiling: in the angles of the rhombi, in the ratio of their areas and in the ratio of their numbers, which approaches φ = 1.618 … with every step. The diffraction pattern of real quasicrystals shows the same ratios. The number that creates the endless self-similarity in the pentagram (see The pentagram and the golden ratio) here orders the atoms of an alloy.
An old forerunner
Long before Penrose, people worked with such patterns. In his Harmonices Mundi of 1619 Kepler drew surfaces of pentagons, pentagrams and decagons and came astonishingly close to a non-periodic tiling. The physicists Peter Lu and Paul Steinhardt showed in 2007 that mosaics on Islamic buildings such as the Darb-i Imam shrine in Isfahan (1453) form almost perfect quasicrystalline patterns from a few basic shapes, some five hundred years before their mathematical description.
So the five, which allows no regular repetition in the plane or in space, does not create disorder but a different, higher order: never the same and yet by strict law. That is exactly how this site describes the role of five in the distribution of the primes (see The disorder of the primes).