We don’t have any trouble coping with three dimensions – or four at a pinch. The 3D world of solid objects and limitless space is something we accept with scarcely a second thought. Time, the fourth dimension, gets a little trickier. But it’s when we start to explore worlds that embody more – or indeed fewer – dimensions that things get really tough.
These exotic worlds might be daunting, but they matter. String theory, our best guess yet at a theory of everything, doesn’t seem to work with fewer than 10 dimensions. Some strange and useful properties of solids, such as superconductivity, are best explained using theories in two, one or even no dimensions at all. Prepare your mind for boggling as we explore the how, why and where of dimensions, starting with possibly the trickiest question of all (see below)…
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3D: We’re here because we’re here?
What is a dimension?
With such a basic question, you might think we’d have a simple answer. Sadly, we haven’t. Defining just what a dimension is turns out to be a surprisingly slippery problem.
The most intuitive description is the oldest one: the number of dimensions a system possesses is the number of independent directions you or anything else can move in. Up and down count as only one dimension because up-ness and down-ness are two sides of the same coin: the further up you go, the less down you are. The same connection exists between left and right, and forwards and backwards, but not between up and right, down and backwards, and so on. Thus, the geometers of Ancient Greece recognised, we live in a three-dimensional world.
So far, so simple, but then things start to unravel. Our place in the cosmos is defined as much by time as it is by space. As long ago as the late 18th century, the Frenchmen and recognised that the mathematical language needed to address time was very similar to that which described space. Time, the mathematicians of the day , was a fourth dimension.
That opened the floodgates. Once untethered from its origins in physical space, the concept of a dimension began to lose its focus. It came to be used as a general term to describe the number of independent coordinates or variables needed to determine the state of any object.
This sleight of hand allowed mathematicians to apply the powerful tools of geometrical analysis to pretty much whatever they wanted. These days an economist, for example, might think of an entire economy as a massively multidimensional object. The price of bread or butter can slide up and down a scale just as we can slide backwards and forwards in spatial dimensions. They are just two dimensions of many millions that describe an economy’s state.
Richard Webb
Seeing in dimensions
Look at the full stop at the end of this sentence. Congratulations: you have just visualised zero-dimensional space. Now run your finger down the edge of the page, then scan it as a whole. That’s one and two dimensions – also easy. But now try thinking of more than three.
Got a headache? If so, you are in good company. “I personally can’t picture more than three space dimensions,” says string theorist of Imperial College London, whose job regularly takes him into 10 or 11 dimensions. A shocking admission, you might think. How can he and his fellow theorists have any confidence that their ideas work?
The answer lies in the work of the 17th-century French mathematician , who showed how the real spaces of geometry could be converted into abstract algebraic equations. Given a line of certain length, for example, you can devise an equation that tells you what x and y coordinates the ends of the line will pass through if it is spun round. This is a circle described in mathematical form.
The idea is powerful because it can be extended into as many dimensions as you like simply by adding more coordinates. In three dimensions, a sphere is described by an equation just like the one for a circle, but with an added set of z coordinates.
So why not carry on from there, and write down an equation for a four, five or six-dimensional “hypersphere”? In 1854, the German mathematician took this bold step, generalising three-dimensional geometry to arbitrarily many dimensions. It turned out to be not such a big deal. “The results aren’t much harder to work with,” says string theorist Edward Witten of the Institute for Advanced Study in Princeton.
Yes, but what do those “high-D” objects really look like? Physicist of New York University says it doesn’t really matter, as long as you come up with some kind of mental picture that works. “The essence of an equation is much easier to store in the brain in terms of images and movies,” he says. For him, Newton’s law of gravity is about a massive object with gravitational field lines spreading out to infinity in all directions. That image works no matter how many dimensions you are thinking of. “The picture is nothing to do with real extra-dimensional space,” Dvali admits, “but it makes it easy to generalise the law to higher dimensions.”
Valerie Jamieson
