Topology explained – and why you’re a donut

You are a donut. Let me explain – it boils down to topology, or the maths of shapeshifting, stretching and pulling, but not tearing.

It makes strange connections between apparently unrelated objects, such as you and the donut.

It also explains exotic quantum behaviour in superfluids and superconductors. And it’s leading to strange new kinds of materials which scientists never thought possible.

Meanwhile, some physicists think topology is the key to the ultimate theory of the universe.

It was for these reasons that topology research bagged the Nobel Prize in Physics this week. 

But what exactly is topology? And where does the physics fit in? Here’s what you need to know.

(W)hole numbers

Topologists define objects according to certain unchanging properties, such as the number of holes something might have.

In topology terms, a sphere is identical to a cube. They are both items with zero holes.

As the mathematics joke goes, a topologist is a person who can’t tell the difference between a donut and a coffee cup – they both have one hole.

Importantly, you can only have an integer number of holes (such as zero, one or two). You can’t have half a hole or a third of a hole. (It’s gotta be a ‘whole’ number, geddit?)

The number of holes in a thing is called a ‘topological invariant’. That just means you can’t change it without tearing.

Though these ideas started out in mathematics, lots of physics systems have analogous ‘invariant’ properties. Topological ideas led physicists to a whole new way to think about matter and the universe.

Basically, some stuff behaves a certain way because of how it is shaped, such as …

Superfluids

While water gushes freely from a faucet, honey slowly drip-drips from a spoon. What defines how easily a liquid can flow is called viscosity and it comes from how the liquid molecules bump into and tangle with one another.

But a superfluid is a liquid that has zero viscosity – it can flow with no resistance at all.

Liquid helium behaves this way when cooled to a couple of degrees above absolute zero (absolute zero is -273 °C). Its atoms adopt a kind of groupthink, acting as if they were a single object.

The result is you can stir a pot of liquid helium and the whirlpool would spin forever without slowing. 

In the 1970s, two of this year’s physics Nobel winners, David Thouless and Michael Kosterlitz, realised that in thin puddles of a superfluid tiny whirlpools can pair up if they spin in opposite directions, like two meshing gears.

But above a critical temperature the two vortices break free from one another and sail away. At this temperature, the liquid helium suddenly changes its behaviour – it undergoes a topological phase transition.

A phase transition happens when something changes its properties, such as water freezing to ice, or your normally docile toddler suddenly throwing a tantrum.

Physicists have used these ideas in lots of other physics research. There is even a quantum theory of gravity (a possible ‘theory of everything’) that treats the fundamental vacuum as a kind of superfluid.

Quantum Hall effect

Usually, the relationship between a current and voltage is a property of a material known as resistance. That’s like the friction stopping electrons flowing through it.

In the 1980s, physicists found a baffling effect in flat materials called the quantum Hall effect where this simple relationship was totally broken.

Instead, the resistance jumped in steps. And, no matter what material was used, these steps were identical.

Topology explained the jumps as coming from the electrons in the material ganging together to form a quantum fluid. Like the superfluid, the behaviour of these electrons are defined only by shape, not by the material they are moving through.

Superconductivity 

Weirder still are superconductors which carry current without any resistance at all. Just as superfluids can have ever-spinning whirlpools, superconductors can have ever-spinning currents.

These loops can be fashioned into quantum bits, or qubits, the building blocks of quantum computers.

Just like the twin whirlpools linking in a superfluid, electrons can pair up in a superconductor, and this helps them glide through on a free ride. Again, topology is crucial to understanding this behaviour.

Topological insulators

These are one of the weirdest kinds of materials discovered in recent years: they are both a conductor and an insulator.

Inside the material, electrons are trapped in local positions, but on the outer surface the electrons can move relatively freely.

Again, this contradictory behaviour arises from shape, and is explained by topology.

String theory

In the esoteric world of superstring theory, physicists attempt to explain all particles and all forces as arising from the vibration of tiny strings much smaller than any known particle. And topology plays a major role.

The strings can be different shapes, not just simple loops. These shapes define what kind of vibrations each string can undergo – kind of like the different notes of a guitar string held at different positions. And the notes define a string’s properties, whether it’s an electron or a quark.

The theory imagines a particular microscopic structure of spacetime with at least six extra dimensions of tangled together at each point all around us.

The way the dimensions are wrapped up is a topological problem.

For instance, the number of holes determines how many families of vibrations a string can feel, and thus how many families of particles exist. Since there are three families of particles, string theorists say the universe must made of three-holed space-time.

Back to breakfast

A donut has one continuous hole through it, and so do you (it starts at your mouth and finishes at the other end).

That’s why, topologically speaking, you and a donut are identical.