Finding the quasicrystal

I read a few articles today that brought aesthetic and religious expression, mathematical curiosity, and physical discovery into contact.

A recent Physics World article reported that an architectural researcher found the first examples of perfect quasicrystal patterns in Islamic architecture.  Also known as Penrose tiles, these patterns were described mathematically by Roger Penrose in the mid-1970s.

Quasicrystals are patterns that fill all of a space but do not have the translational symmetry that is characteristic of true crystals. In two dimensions this means that sliding an exact copy of the pattern over itself will never produce an exact match, though rotating the copy will often produce a match. They were first described mathematically by the British academic Roger Penrose in the guise of the famous Penrose tiles. About 10 years later Danny Schechtman of Israel’s Technion University showed that the positions of atoms in a metallic alloy had a quasicrystalline structure. Since then, hundreds of different quasicrystals have been discovered in nature.

In 2007 two physicists in the US reported that they had found an example of a 15th-century geometric pattern in Iran that showed an “almost perfect” example of Penrose tiling. These researchers concluded that the Islamic craftsmen most likely created the patterns using a set of tiles of distinct shapes, each decorated with lines that join to form the final patterns. Several other studies have also suggested that quasiperiodic patterns in Islamic architecture were constructed through local rules such as subdividing or overlapping of tiles. But none of the proposed methods has able to explain how the ancients ended up creating global long-range order in their patterns.

But the new paper identifies three perfect examples of these patterns (one from as early as the 12th century). It also makes an observation that could clarify how they were created.

In this latest work, Rima Ajlouni, an architectural researcher at Texas Tech University in the US, believes that she has identified three examples of quasiperiodic patterns in Islamic architecture without any imperfections.

In her paper, Ajlouni also shows that ancient Muslim designers were able to resolve the complicated long-range principles of quasicrystalline formations. In other words, these designers were fully aware of the extent of connectedness within their work. In all three examples, Ajlouni reconstructs the patterns and shows that the size of a central “seed” figure is proportional to the size of the overall framework of the pattern. She demonstrates that the three patterns could have been created using nothing more than a compass and a straightedge. This construction method that was widespread in Islamic societies to create a variety of media such as woodworks, ceramics and tapestries.

“They were able to create some of the patterns of complex modern mathematics using basic principles alone,” she says.

Some of the inspiration for Penrose’s work came from Kepler’s Monsters. And a New Scientist article on Shechtman’s prize also identifies the quasicrystal’s relationship to another well know mathematical idea:

The quasicrystal is related to the Fibonacci sequence, in which each number is the sum of the two preceding it, such as 1, 1, 2, 3 and so on. The terms in this sequence yield the “golden ratio” tau, and it turns out that tau governs the distances between atoms in quasicrystals.

Another PhysicsWorld article reporting Shechtman’s prize, includes some nice detail.

Finally, an October article in +Plus Magazine explores the relationship between Penrose tiling and Shechtman’s discovery of a quasicrystalline structure.

Penrose was aware of the potential connection to crystallography when he discovered his first non-periodic tilings which had five-fold symmetry in the early 1970s – and that a lattice of atoms following his non-peridoc pattern would disobey the standard rules of crystallography. But creating a Penrose tiling is not straight-forward: most people on their first attempts get stuck with a configuration that can’t be extended any further.

“For this reason I was somewhat doubtful that nature would actually produce such ‘quasi-crystalline’ structures spontaneously,” says Penrose. “I couldn’t see how nature could do it because the assembly requires non-local knowledge. With my tilings you can’t hope to get it right over a large area simply by placing the atoms one after the other where they ‘fit’. At various times a choice would arise and to make the right choice you may have to know what’s going on a long way away at the other end of the structure. If you make the wrong choice you eventually get stuck.”

But it seemed that nature had found a way. The chemist Alan Mackay had used Penrose’s tiles to build a model of a two-dimensional crystal – a layer of atoms sitting at the corners of a Penrose Tiling. This non-periodic crystal would have five-fold symmetry and its diffraction pattern would have the same features as those produced from Shechtman’s crystal. Roger Penrose’s non-periodic tilings of two-dimensions, and their three-dimensional analogues, explained the structure of Shechtman’s impossible crystal.

“Schechtman’s discovery did come as a bit of a surprise to me,” says Penrose. “But as soon as Paul Steinhardt [one of the physicists responsible for making the connection between Mackay’s model and Shechtman’s diffraction pattern] showed me the diffraction patterns that Shechtman had found and that he (Steinhardt) had calculated I was happy to believe that nature had found a way around the problem.”

The mathematics that Penrose investigated gave structure to the raw observation first made by Schechtman.  It is the diffraction patterns of an X-ray beam (scattered by the atoms of a substance that was put in its path) that chemists use to construct the three-dimensional structure of a crystalline substance. They look at the symmetries in the diffraction patterns.  And the mathematics of group theory provides the language they need.

The 12th century version of Penrose’s tiling idea certainly tells us that there is more than one path to an insight.  But the mathematics of this insight, applied to the Schechtman’s observation, is what makes his observation comprehendible.  I inevitably think that the role  mathematics is playing resembles the nervous system itself, finding structure in data.  Daily, the nervous system’s sensory and cognitive processes bring structure to the sensory data that flood the body.

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