Alice and Bob are colleagues at a New York law firm. They are also lovers. They devise a secret tryst at a little-known hotel in Manhattan. “It’s on the corner of 58th Street and 12th Avenue,” whispers Alice as they stand by the office photocopier. “I’ll meet you by the elevators on the 8th floor.” Bob smiles and nods.

Alice has specified the location of their meeting by assigning three numbers: 58, 12 and 8. Alternatively, (had the two lovers been geography professors, perhaps) Alice could have used latitude, longitude and altitude. The point is that with three numbers, we can uniquely fix any given point in space, and for that reason we say that space is three-dimensional.

As Bob turns away he realises there’s one thing left to settle. “What time, Alice?”

“Tonight, 7 pm,” she replies.

By adding a fourth number (7, or 19 for those who prefer a 24-hour clock), Bob and Alice’s tryst is a planned event in space *and* time, what physicists call spacetime.

Physicists said spacetime has four dimensions. But as with many things in physics, this is not quite as simple as it seems. The three-dimensional nature of space is so obvious and intuitive that most people – even scientists – never stop to think about it.

But in 1921, a German mathematician named Theodor Kaluza proposed that our intuition had misled us: he suggested that space really has four dimensions, and therefore spacetime is five-dimensional rather than four‑dimensional. He arrived at this bizarre conclusion by discovering an amazing mathematical fact while playing with the period’s most important piece of physics: Albert Einstein’s general theory of relativity.

Kaluza had taken the equations of Einstein’s theory, which were formulated to apply to the familiar four spacetime dimensions, and rewritten them to apply to five dimensions instead. Why? Well, it’s the sort of thing mathematicians do. But the result was stunning. Viewed from the normal four-dimensional perspective, Kaluza’s equations reduced to those of Einstein’s theory, but with an extra set of terms (describing the extra dimension).

Surprisingly, these terms corresponded precisely to the description of electromagnetism that James Clerk Maxwell had published decades before. By adding an extra dimension of space, Kaluza had, it seemed, accidentally unified gravitation and electromagnetism, two of the fundamental forces of nature.

There was only one snag with this five‑dimension theory: where is the extra space dimension? We don’t see it. How could we have overlooked something so basic? An answer came a few years later from a Swedish physicist, Oskar Klein. Maybe, thought Klein, we don’t notice the fourth dimension of space because it is rolled up to a very small size.

To understand what this means, imagine viewing a drinking straw side-on, from a distance. It looks like a one-dimensional line. Only on closer inspection do we see the line is really a tube. Any given point on the ‘line’ is actually a little circle that’s part of the tube. Klein claimed that’s where Kaluza’s extra dimension was hiding – that what we normally consider to be dimensionless points in space are in reality tiny circles, adding a fourth dimension too small to see or even notice in experiments.

Kaluza and Klein’s theory remained a mathematical curiosity for some decades, but in the 1970s some physicists began to wonder whether their idea could be extended from just gravity and electromagnetism to include the two additional forces of nature: the weak and strong nuclear forces. Sure enough, it can, although to incorporate these more complicated forces we have to add not just one, but six extra dimensions, making 10 space dimensions – 11 dimensions in total, if you count time.

Just as in Klein’s original proposal, the extra dimensions could be rolled up – “compactified” is the technical term – to a tiny size, much smaller than an atomic nucleus. Unlike a single extra dimension, which can only be compactified into a circle, there are now choices: for example, two dimensions can be compactified into either a sphere or a torus (doughnut shape). For six dimensions, the number of combinations rockets. The particular way the dimensions are compactified affects the properties of the various forces.

The basic idea of unseen extra dimensions has found an enthusiastic reception among string theorists. String theory purports to describe all the forces and particles of nature in terms of little loops of string that wriggle around in higher-dimensional spacetime, with 11 spacetime dimensions a leading contender. String theory is currently the front-runner for a unified theory of fundamental physics.

It took a long while for the idea that spacetime has more than four dimensions to be accepted as a model for real space, but today it is considered the default option. If the idea is right, then compactification is not only critical for physics, but for lovers too. Most of these dimensions are rolled up so small we don’t have to worry about them in our everyday lives. So we only have to remember four dimensions, not 11 or more, to keep a rendezvous!