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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…

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 Jean le Rond d’Alembert and Joseph-Louis Lagrange recognised that the mathematical language needed to address time was very similar to that which described space. Time, the mathematicians of the day rapidly came to agree, 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 Michael Duff 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 René Descartes, 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 Bernhard Riemann 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 Gia Dvali 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

0D – On the dot

The idea of something having zero dimensions carries a whiff of the emperor’s new clothes. Surely, with no dimensions there’s no room for anything, so a 0D space must amount to nothing at all – mustn’t it?

Not necessarily. Some of the hottest properties in physics are 0D semiconductor structures known as quantum dots. Anything from nanometres to micrometres across, they do admittedly have a size, but electrons can be crammed into them so tightly that they have no dimensions to move in at all.

“It’s a zero-dimensional trap for charges,” says Leo Kouwenhoven of Delft University of Technology in the Netherlands. Electrons confined in this way start to behave very strangely indeed, and quite usefully.

For a start, any energy you pump into a quantum dot cannot be used to shuffle the electrons around, but can only be released as light. This makes quantum dots promising as a highly efficient low-power light source. Because they are so small, the dots might also serve as fluorescent markers to label biological molecules such as antibodies in order to track their progress through a living organism.

Kouwenhoven admits that this is still some way off; first we’ll have to fabricate quantum dots from proven non-toxic materials, he says. His own research focuses on another potentially hot application. Each excited electron trapped in a quantum dot produces exactly one photon, and so information can be transferred reliably back and forth between the electron and photon. That could make quantum dots just the right medium in which to manipulate and store data in a first generation of quantum computers – the awesomely powerful devices which, if one big enough can ever be built, promise to transform how we process information.

“We’ll have a proof of principle probably a few years down the line,” says Kouwenhoven, “and maybe commercial applications in a decade or so.” Encouragement enough, perhaps, that it is not always nothing that comes of nothing. Richard Webb

1D – Walk the line

One is the dimension in which physics starts to look a little familiar. A single dimension is just a straight line, the perfect environment for the operation of such classical concepts as Newton’s laws of motion.

But it is in quantum theory that 1D physics really comes to life. “You get completely novel effects – things that you don’t get in any other dimensionality,” says Thierry Giamarchi, a specialist in 1D materials at the University of Geneva in Switzerland.

Take the behaviour of electrons. They will normally do anything to keep out of each other’s way, but trapped in a 1D channel where they can only move backwards and forwards they begin to interact and move as one. Get the conditions right, though, and things go the other way: a confined electron can be made to act as if it were two particles, one with the electron’s charge, and one with its spin. “There are a host of phenomena like this,” says Giamarchi.

Such tricks are the delight of physicists, but their significance goes way beyond that. As electronic devices get ever smaller, 1D will increasingly become the place to be. 1D carbon nanotubes can be made fully conducting or semiconducting at will, and are a hot prospect for wiring up future generations of computer chips. Michael Brooks

1½D – Fractal landscapes

We live in a world of three-dimensional objects bounded by two-dimensional surfaces and outlined by one-dimensional lines. All in all, a comforting, intelligible, whole-number sort of world.

Or do we? As the mathematician Benoit Mandelbrot pointed out in his 1982 book The Fractal Geometry of Nature, clouds are not spheres, mountains are not cones and coastlines are not circles. The dimensions of the raw, rough real world do not, it turns out, come in tidy integers.

Imagine, for example, tracing the delicate outline of a snowflake. As you zoom in, you find yourself following an ever more intricate pattern, and the closer you get, the longer the line you trace becomes. Your drawing is still a line, but its crinkles embrace far more of the space on the page than a straight line. And yet a line, however hard it squirms, can never be more than a 1D object. Or can it?

Welcome to fractal dimensions, the irregular landscapes between the familiar worlds of one, two and three dimensions. Fractal dimensions are not the same as the left-right, back-front and up-down directions we’re used to, but they are intimately related: they describe how much more space a complex object fills as you look it at finer scales and measure more of its detail (see diagram).

It’s not just about snowflakes. Many natural objects have fractal geometry: river networks, branching lightning, clouds, broccoli. You might even claim to live in a fractal landscape, more or less so depending on where you are in the world. The rugged coastline of Great Britain, for example, differs in length wildly according to whether you measure it with a yardstick or calipers. It has been calculated to have a fractal dimension of about 1.25. Smooth South Africa, on the other hand, is only slightly rougher than a straight line, with a fractal dimension of 1.02.

Valerie Jamieson

2D – Vistas of flatland

“Two dimensions are golden,” says Andre Geim of the University of Manchester, UK. Physics in one dimension is too simple to be satisfying, and three dimensions are complicated and messy. Two-dimensional “flatland” is just right, with just enough room for interesting and useful things to arise. “As a physicist, this is the dimension you would like to live in,” says Geim.

He would say that. Geim was one of the team that in 2004 produced the first 2D material, sheets of carbon one atom thick known as graphene. Graphene could indeed be incredibly useful, with electrons shooting across its sheets almost unhindered. If 1D nanotubes are the wires of future computers, graphene could be their circuit boards.

That’s not all. Take high-temperature superconductors. We already know of materials that conduct with absolutely no resistance at temperatures up to around 130 kelvin, just under halfway from absolute zero to room temperature. We’d love to know how they do it, but after 20 years of head-scratching all we know is that the effect seems to arise from the formation of 2D “stripes” of electrical charges. If we could fully fathom the physics behind that, it could set us on the path to superconductors that work even at room temperature.

So flatland is practical, but it is also profound. When electrons are confined by powerful magnetic fields to a 2D layer of semiconducting material cooled to less than one-third of a degree above absolute zero, electrons – which were long considered fundamental, indivisible particles – appear to break down into particles each with just a fraction of the electron’s charge.This phenomenon is known as the fractional quantum Hall effect, and the resulting particles are ambiguous characters dubbed anyons.

Anyons not only force us to rethink the nature of the electron, but might also, like zero-dimensional quantum dots, represent a great hope for building an ultrapowerful quantum computer. If we could get such a machine to work on a significant scale, it would perform astonishing feats of information processing, and could also faithfully model the behaviour of quantum systems. In short, flatland may open grand vistas on everything from new drugs to parallel universes. Michael Brooks

3D – We’re here because we’re here?

2D flatland and multi-dimensional hyperspace make fine playgrounds for the mind, but our bodies seem stuck in a space of three dimensions. Why not two or four, or five or more? Recently, physicists trying to meld gravity and quantum theory, and so explain the nature of space and time, have begun to revisit this old question.

String theory, one route to quantum gravity, gives an unsatisfactorily vague answer: space can have anything from zero to 10 dimensions. That drives theorists to anthropic arguments: universes of all possible dimensionalities exist, but we see what we see because beings like us require a 3D habitat.

In 2005, Andreas Karch of the University of Washington, Seattle, and Lisa Randall of Harvard University came up with a more mechanistic explanation of the mystery of threeness. They created a model in which many universes of different dimensions float around inside an expanding 10-dimensional hyperspace of the kind popular in string theory. When these universes collide, they annihilate one another. The calculations showed that three and seven-dimensional universes are the ones most likely to survive such catastrophes.

If you accept the premise, that almost answers the question – but why shouldn’t we live in a spacious realm of seven dimensions instead of our cramped 3D universe?

That might be explained by looking at space not as a uniform whole, but as a construction built up from tiny pieces. A European team has done just that with higher-dimensional analogues of the triangle, the simplest unit that can be stuck together in different ways to make curvy universes. Quantum theory says that the “true” shape of the cosmos should be the sum of all these possibilities. By requiring that their model universe should adhere to strict cause and effect, the team found that the result has just one dimension of time and exactly three of space.

There is a twist. At the very smallest scales, the structure of space changes: one dimension of the three melts away to leave only two dimensions. Perhaps, if you look closely enough, we live in flatland after all. Stephen Battersby

4D – Time, the great deceiver

Space consists of three dimensions. Time, we are told, is also a dimension. So how come it is so different?

Answer: it isn’t. “Space and time are not concepts that can be considered independently of one another,” says physicist Roger Penrose in his book Gravitation. In Einstein’s special theory of relativity, they dissolve into one entity. Two objects that to one person seem separated only in space can to another seem divided by space and time. Similarly, two events that seem to be separated only in time might from another perspective occur in different places as well.

That does not chime with our everyday experience, but only because we are not up to speed. Discrepancies between two observers’ views only become obvious when their relative speed is close to the speed of light – the cosmic speed limit.

Einstein’s physics reveals a deep truth: space and time are just different threads of a single seamless fabric called space-time. Yet there is still an obvious difference between the two. We can, in principle, travel in any direction in the three dimensions of space, but we can plod in one direction only in time – forwards from the past to the future. How do we explain that anomaly?

Ultimately, says physicist Lawrence Schulman of Clarkson University in New York, that too is down to the cosmic speed limit. Imagine, for example, pulling back your curtains at 7 am on a bright, sunny morning. The sun, though, already doesn’t exist. It exploded at 6.55 am; we just don’t know it yet, because light from it takes about 8 ½ minutes to travel to Earth.

In this picture, any event – the exploding sun, us standing at the window – is a point in space-time with two associated “light cones”. One represents light racing towards the event through space and time, and one represents light racing away (see diagram). To see the sun explode as it happens would mean stepping outside our light cone and moving faster than the speed of light – something that our universe does not allow.

“It is the cosmic speed limit that makes some parts of space-time inaccessible,” says Schulman. It breaks the symmetry between time and space – and means we see information flowing smoothly in what we call time from the past to the future. Marcus Chown

5D – Into the unseen

“The question ‘how many dimensions does the universe have?’ may not have a unique answer”

Seeing time as the fourth dimension made sense of Einstein’s special relativity. The German mathematician Theodor Kaluza had even grander designs. In 1919, he sent a paper to Einstein in which he argued that by adding a fifth dimension to space-time, it was possible to show that gravity and electromagnetism were two aspects of one and the same force.

A few years later a Swedish mathematician, Oskar Klein, took Kaluza’s idea and ran with it. He countered the obvious objection to the existence of a fifth dimension – that it is not immediately apparent – by showing it could be tiny and curled up at every point in our four-dimensional space-time. Thus began the trend of searching for the unification of forces in the hidden dimensions of hyperspace that continues today in string theory.

Perhaps, however, the fifth dimension is not as tiny as Klein suggested. Using string theory, Harvard physicist Lisa Randall and Raman Sundrum of Johns Hopkins University in Baltimore, Maryland, showed in 1999 that a fifth dimension could explain a vexing mystery: why gravity seems so much weaker than nature’s other forces. Their model has our familiar four dimensions floating in an infinitely large, negatively curved fifth dimension. While the electromagnetic and nuclear forces are stuck inside a “brane” made of four dimensions, gravity leaks out into the fifth.

Meanwhile, Paul Wesson of the University of Waterloo in Ontario, Canada, has argued that reality in fact has five dimensions that can be broken down into our four familiar dimensions plus the mass that populates our world. This theory not only rids physics of the problem of why things have mass – it becomes an incarnation of geometry – but also of the big bang singularity, the vexing state of infinite temperature and density from which our universe sprang and where all current physical theories break down. Seen from the perspective of the full 5D universe, the big bang beginning is no more than an illusion.

The possibility of five dimensions is also raising more subtle questions. In one of the most remarkable results of string theory, theoretical physicist Juan Maldacena posited in 1997 that some string theories in five large dimensions that include gravity are equivalent to ordinary quantum-field theories in four dimensions without gravity. The former is a holographic projection of the latter – potentially making our everyday world as ethereal as a hologram projected from the boundary of the universe.

That might sound esoteric, but the correspondence has recently been successfully applied to computationally difficult problems in many areas, including the physics of high-temperature superconductors. In Maldacena’s picture, the 4D theory is no “truer” a description of the world than the 5D one. Seen in that light, the question, “how many dimensions does the universe have?” does not have a unique answer after all. Amanda Gefter

6D – Two-timing

Whenever physicists devise theories of the universe that invoke extra dimensions, they always seem to mean the kind in which you could theoretically, if you could find them, move freely. In other words, hyperspace is just that: space. When it comes to higher dimensions, time does not get a look in.

There’s a good reason for that. If there were more time-like dimensions, things could shuttle between arbitrary points in our one-dimensional time by looping through other time dimensions, circumventing the limits imposed by light’s finite speed. In other words, time travel would be possible. In our cosmos at least, that seems not to be the case.

In 1995, though, Itzhak Bars of the University of Southern California in Los Angeles saw a hint of an extra time dimension in M-theory, the umbrella version of string theory. Investigating further, he used various tricks to construct a theoretical framework in which a second time dimension could exist, but where time travel was not allowed. Such a two-timing theory would have attractions, as Bars’s subsequent research has shown. It might, for example, iron out some wrinkles in the standard model of particle physics.

The catch is that the scheme only works if there is an extra spatial dimension, too. Yet Bars found that things in this 6D universe would look pretty much like they do in our 4D universe – with one difference. There will be not one standard model of particle physics, but many 4D “shadows” of a 6D original. Marcus Chown

8D – Surfer’s paradise

Eight dimensions is a rarefied space that is home to the octonions – “the crazy old uncle nobody lets out of the attic”, as mathematician John Baez of the University of California, Riverside, puts it.

Octonions are indeed odd creatures. They are one of only four number systems in which division is possible, and so allow the full range of algebraic operations to be performed. The way octonions interact, however, is peculiarly exasperating and unlike anything we are familiar with from our conventional number system (see diagram).

So why bother with octonions at all? It’s because, for some problems in theoretical physics, they are an invaluable tool. Matrices filled with octonions are the building blocks of a bizarre mathematical structure known as the “E8 exceptional Lie group“, which sits at the heart of a particular form of string theory.

In 2007, E8 hit the headlines when physicist Garrett Lisi, who has no university affiliation and spends most of his time surfing in Hawaii, used the E8 group to seemingly unify gravity with the three other fundamental forces without using string theory. The publicity surrounding that work ruffled some feathers. “String theorists have been working on [E8] since the late 70s,” says Michael Duff of Imperial College London. “We didn’t need surfer dudes to tell us that it was interesting.”

Duff himself is agnostic about the value of octonions, pointing out that no theory in which they pop up has yet been tested by experiment. “Whether octonions have anything to do with the real world is still anybody’s guess,” he says. Anil Ananthaswamy

10D – String country

“Perhaps the most outrageous idea that physicists have brought back from higher dimensions is that all possible universes exist”

Ten dimensions, and we finally reach the fabled land of string theory. For all the vitriol that has been thrown at it, string theory is for the moment the only real game in town when it comes to attempts to bundle up quantum mechanics and general relativity into a “theory of everything”. It holds that all particles that make up matter or transmit forces arise from the vibration of tiny strings. Those strings are one-dimensional. The space they wiggle about in is not. In fact, it has 10 dimensions: nine of space, and one of time.

Why? In a nutshell, because the theory doesn’t work with any fewer, as physicists Michael Green and John Schwarz showed in 1984: mathematical anomalies crop up that translate into violent fluctuations in the fabric of space-time at scales smaller than the Planck length of 10^-35 metres.

That doesn’t necessarily mean that 10 is the magic number. Indeed, one now unfashionable early variant of string theory had 26 dimensions. There are five broadly defined brands of 10D string theory that compete to explain the universe, with no indication as to which, if any, is the right one. But these disparate theories can be unified into one overarching theory, known as M-theory. M-theory has 11 dimensions.

It is assumed that the extra dimensions of M-theory must in some way be squashed down to a size that we can’t see. The bad news is that there is an almost unlimited number of ways in which this can be done. How to single out the one way that produces our universe remains a problem. “It divides theorists into two camps,” says Michael Duff of Imperial College, London. Those who say we’ll work out the trick eventually are faced by a growing band who subscribe to the alternative view of the “multiverse”. This is perhaps the most outrageous idea that physicists have brought back from their expeditions into higher dimensions: that all possible universes do actually exist. The universe we know is as it is because it just happens to be the one we are living in. Anil Ananthaswamy ■

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