Editorial: “The deep value of mathematics”

WE ALREADY have solid, liquid, gas, plasma and Bose-Einstein condensate. Now it seems we may be on the verge of discovering a whole host of new forms of matter – all based on mathematics.

Nils Baas, a mathematician at the Norwegian University of Science and Technology in Trondheim, has unearthed a plethora of possibilities for the way the components of matter can link together.

He made the discoveries while researching the field of topology – the study of the properties that objects share because of their shape.


It is particularly concerned with the various shapes that can be formed while squashing and bending an object. A ring doughnut and a teacup share the same topology, for example: it is possible to squish the doughnut into a teacup shape without doing away with the hole, as it becomes the hole in the handle.

Baas was studying “Brunnian rings” – collections of rings that are linked together but can all be separated if only one ring is cut. Borromean rings are the most famous example (see illustration).

Baas has shown that many more linkings are possible: not only are there Brunnian links of four or more components, there are also sets of Brunnian links which are themselves linked together in a Brunnian fashion to create what Baas calls “hyperstructures” (arxiv.org/abs/1012.2698).

In 1970, Vitaly Efimov, now at the University of Washington in Seattle, predicted that the topology of the Borromean rings would be reflected in nature as a hitherto-undiscovered form of binding between three particles. In the last five years, it has been shown that some of these links can indeed occur in physical systems. In 2006, researchers found this “Efimov state” in a gas of ultra-cold caesium atoms: each atom had a single link to one of the others, but picking up one moved all three (Nature, DOI: 10.1038/nature04626).

Then, in 2010, Japanese researchers found Borromean rings in the bonds between atomic nuclei (Physical Review Letters, vol 104, p 062701). “These structures seem to act as a recipe for what you can construct in the real world,” Baas says.

These structures seem to act as a recipe for what you can construct in the real world

But Baas’s more complicated hyperstructures have radically different topologies from anything yet seen in nature. If groups of particles can be made to bond in this way, they would create matter with previously unseen properties, Baas reckons. “When you go to a higher level, something completely new happens mathematically – and I would suspect it does in the real world too,” he says.

Baas has teamed up with Ned Seeman of New York University in New York City to figure out how to build the hyperstructures. “Mathematics seems to be a pretty good predictor of reality,” says Seeman, who synthesised Borromean rings using DNA strands in 1997. “I have every suspicion that they’ll work out.”

Baas has plenty of other avenues to explore, too, including a new inroad into the fundamentals of quantum theory. Particles that interact can become curiously synchronised, even when separated, in a quantum process called entanglement. If the particles start out linked together in complex Brunnian ways, they might be able to affect each other even when separated, he says, providing new ways to create the spooky-action-at-a-distance connections like those observed in entangled systems.

“Once [these links] have been pointed out from topology, then we can go back and look for them in the Schrödinger equation” that describes the mathematics of quantum theory, says Baas.

From there, it might be possible to create new quantum states in the lab. This in turn might provide new ways to build super-powerful quantum computers, which manipulate information carried in the quantum states of particles. Such quantum information can be in many states at once, so quantum computers can carry out enormous numbers of calculations simultaneously.

The original version of this story incorrectly stated that each of the three Borromean rings is threaded through only one other. That has since been fixed.