What’s up with all these dimensions?

In Lewis Carroll’s Through the Looking Glass and What Alice Found There, Alice steps through a mirror into a strange alternate dimension. In this reflected universe, she journeys across a ‘curious country’, a landscape marked out as a giant chessboard. Over the course of the story, she advances square-by-square across this two-dimensional board game before arriving at the final square and being crowned queen.

The idea of travelling to an alternate dimension has inspired many fictional worlds, from Alice’s adventures to Stranger Thing’s Upside Down and Miles Morales’ many Spider-verses. For physicists, however, dimensions aren’t separate universes that humans can enter. Instead, dimensions are measurements defining a point in spacetime. Playing with dimensions – whether it’s performing calculations with more dimensions or less - offers physicists like Perimeter’s research faculty Pedro Vieira and resident PhD student Raquel Izquierdo Garcia new ways of understanding our universe.  

What is a dimension? 

Mathematically, dimensions are defined by how many numbers you need to identify a point.

Imagine you’re a little stick figure, and your whole world and everything you’ve ever known exists on a single line. You can move left and right along the line – but that’s it. You would only need one single number to identify where you are on that line. You could be at marker 1, 2, or 3, for example. This is a one-dimensional world.

Now one day your world expands – you find yourself in a two-dimensional world. You can move left, right, forward, and back – as if you’re travelling across Alice’s chessboard. You need two numbers to identify your location, a latitude and longitude if you will.  

Jump out of the boardgame and into the real world, and you’re going to need an extra number to define your location. Not only can we move left, right, forward, and back, but we’ve now added up and down into the mix. You need three numbers – latitude, longitude, and altitude, to pinpoint where you are in space.  

Let’s throw one more dimension into the mix: time. Einstein’s theory of relativity joins space and time into four dimensions.  

“We all understand that we live in a three-dimensional world, because I can move forward, I can move right…but the other thing is that time never stops,” says Raquel Izquierdo Garcia, resident PhD student studying quantum fields and strings. “If I just stay in a chair and don’t make any movement in space, things are going to change because time is evolving.”

But for physicists, three dimensions of space plus a fourth dimension for time aren’t the only game in town. You can also work in fewer or many more dimensions.  

Simplify, simplify

If you’re planning on redecorating, there’s only two dimensions that matter when calculating how much wallpaper to buy: the height and width of the wall. That mindset – that problems can be simplified into fewer dimensions – turns up in physics as well, where it’s referred to as lower dimensions.

In general, researchers study phenomena in lower dimensions because it’s easier, says Pedro Vieira, Clay Riddell Paul Dirac Chair in Theoretical Physics at Perimeter. “One of the main reasons we love lower dimensions is that we can draw pictures,” he says. “It’s much easier to draw a map than to draw something in 3D.” Sometimes, these simplifications even match reality closely.

Vieira points to one example. “When we scatter things like particles at super high energy, it’s very common for objects moving super fast to behave like pancakes,” he says.  

“Effectively, when they bounce against each other and collide, it’s as if there were two very thin pancakes,” he explains. “So at high energy, sometimes what matters is lower dimensional physics.”

Two-dimensional thinking is also important in the study of condensed matter physics and quantum materials, says Izquierdo Garcia. “In the lab, people can make materials that are only one atom high. When you study these kinds of systems, you need to study a two-dimensional world.”

Projecting the universe

Physicists can also use lower dimensions to describe what happens in higher (aka, more) dimensions. One such idea that’s promoting a flurry of research at Perimeter is holography, the theory that gravity in a three-dimensional universe could be described by a hologram in two dimensions.  

“A hologram is a two-dimensional surface that you look at it, and it looks 3D,” explains Vieira. “But the 3D is fake. It’s really 2D – the 3D is an illusion. One theory of physics says that perhaps we are like a hologram – perhaps the mathematics of us can be described with less dimensions in the same way that we describe holograms with less dimensions.”

This theory is informed by research into black holes. People discovered that a black hole’s surface governs what exists inside– so even though inside a black hole there is a full 3D volume, the surface, which is one less dimension than the inside, seems to be what matters.  

“These ideas in black holes led people to think, could this be not just for black holes? Could it be that even beyond black holes, information can be encoded at the boundary?” asks Vieira.  

These questions occupy a host of researchers at Perimeter. In fact, all the Perimeter faculty in the Quantum Field and Strings research area work on holography routinely, probably daily even, says Vieira.  

Honey, I shrunk the dimensions  

Other theories in physics use six, eight, ten or more dimensions. In these scenarios, the universe still appears to look 3D, but that’s just because these extra dimensions are very, very small.  

Izquierdo Garcia says thinking about super small dimensions requires some imagination. She uses an ant as an analogy.  

She holds up a pen so that it resembles a flat line. “I have a pen, and if I look at this pen from far away, I think it is one line.” That perception changes when you zoom in. “If I was a very small ant, I could walk in the transverse direction and discover that this pen is two-dimensional instead of one-dimensional.”  

The physics that Izquierdo Garcia works on is 10-dimensional. “We are studying these string theories defined in 10 dimensions and sometimes I can split them into two: the four large dimensions we live in and six very small dimensions in which we cannot travel. Studying the effects of these extra dimensions gives us a lot of new physics to study!”

Back through the looking glass

Whether you’re reading coordinates on a map or trying to understand the structure of a black hole, dimensions are a key part of mathematical physics. And don’t worry too much if your brain’s a little sore from trying to contemplate a world of extra dimensions. Instead, think of dimensions as a valuable tool we can manipulate to uncover new discoveries about our universe. Whether we’re condensing down for simplicity’s sake or extrapolating up to dive deeper into what’s possible, playing with the number of dimensions in a theory is just another way to explore.