Physicists dream up crystal based on elegant satellites

   A crystal’s calling card is its regular, repeating pattern of unmoving atoms or molecules. But now a team of physicists have proposed a new kind of crystal that gets its order from the carefully composed movement of its components. Called choreographic crystals, these structures may look messy at any given moment, but display symmetric, repeating patterns if you watch them over time.

“Because of the motion in the constituents, you can have a much higher symmetry than is visible in a snapshot,” says Latham Boyle of the Perimeter Institute for Theoretical Physics in Waterloo, Canada.


Boyle came up with the idea while trying to design a future detector for gravitational waves, ripples in space-time that are proposed by Einstein’s general relativity but have not yet been detected, despite recent rumours. One possibility involves a trio of satellites that can detect small changes in the distances between them, which would be caused by a passing gravitational wave. The LISA Pathfinder satellite, launched last month, is a precursor to this sort of observatory.

But because the three satellites always make a triangle, it takes time for them to sweep through space. You need at least four satellites to get an instant picture of the passing wave. “It seemed too futuristic to really work out in gory detail, but I couldn’t resist doing it for the simplest case,” Latham says. “I quickly realised that it was not at all obvious what the answer was.”

Satellite arrangement
He started looking for the most symmetric way to arrange four satellites, where symmetry is defined as the number of ways you can shift or rotate the system and have it still look the same. For four static points, the most symmetric arrangement is a regular tetrahedron, which looks like a triangular pyramid. There are 24 different ways to rotate or reflect this solid body without noticeably changing it.

Unfortunately, once you let the points move in circular or elliptical orbits, there’s no way to maintain that 3D shape. “That was disappointing,” Boyle says. “But then I thought maybe I’m being too rigid about what I’m requiring of the orbit. Maybe it’s symmetric in a more dynamical sense.”

Imagine if you were an astronaut on one of these satellites, and filmed everything you saw out of the spaceship window. If someone else watching the movie later couldn’t tell which satellite you had been on, then that satellite configuration has dynamical symmetry, Boyle says.

Mixing the usual rotation and reflection symmetries with this new symmetry of shifts or reflections in time creates what Boyle and his colleagues call choreographic symmetry.


Defined that way, there are 48 ways to translate the four satellites’ arrangements without changing how they look – more symmetry than for the static scenario. “It turned out to have way more symmetry than I expected it to have, way more than I thought possible,” Boyle says.

“It’s inspiring,” says Frank Wilczek at the Massachusetts Institute of Technology, who proposed the idea of a “time crystal” that has symmetry in time the way ordinary crystals do in space. “It shows you can get some really interesting structures when you start to play these games.”

Appearing naturally
Boyle and his colleagues generalised these “choreographic crystals” to find the analogues of other shapes. They speculate that such things could be built in the lab, perhaps by using optical lattices that trap atoms, or even that they could show up in nature.

There is precedent for this idea: quasicrystals, which have ordered but non-repeating patterns, were first proposed theoretically, then built in the lab (garnering their discoverers a Nobel prize), and finally found naturally in a meteorite that had landed in a remote part of north-eastern Russia.

Ordinary crystals with high degrees of symmetry also have interesting applications in unexpected places, like data compression and string theory. “We have this dream that these choreographic lattices are similar..