© Brian R. Greene 2004 
In 1919, Einstein received a paper that could easily have been dismissed as the ravings of a crank. It was written by a little-known German mathematician named Theodor Kaluza, and in a few brief pages it laid out an approach for unifying the two forces known at the time, gravity and electromagnetism.
To achieve this goal, Kaluza proposed a radical departure from something so basic, so completely taken for granted, that it seemed beyond questioning. He proposed that the universe does not have three space dimensions. Instead, Kaluza asked Einstein and the rest of the physics community to entertain the possibility that the universe has four space dimensions so that, together with time, it has a total of five space-time dimensions.
Kaluza proposed that in addition to left/right, back/forth, and up/down, the universe actually has one more spatial dimension that, for some reason, no one has ever seen. If correct, this would mean that there is another independent direction in which things can move, and therefore that we need to give four pieces of information to specify a precise location in space, and a total of five pieces of information if we also specify a time.
Kaluza realized that the equations of Einstein’s general theory of relativity could fairly easily be extended mathematically to a universe that had one more space dimension. Kaluza undertook this extension and found, naturally enough, that the higher-dimensional version of general relativity not only included Einstein’s original gravity equation but, because of the extra space dimension, also had extra equations.
When Kaluza studied these extra equations, he discovered something extraordinary: the extra equations were none other than the equations Maxwell had discovered in the nineteenth century for describing the electromagnetic field! By imagining a universe with one new space dimension, Kaluza had proposed a solution to what Einstein viewed as one of the most important problems in all physics. Kaluza had found a framework that combined Einstein’s original equations of general relativity with those of Maxwell’s equations of electromagnetism.
Then, in 1926, the Swedish physicist Oskar Klein injected a new twist into Kaluza’s idea, one that suggested where the extra dimension might be hiding. Klein’s contribution was to suggest that what’s true for an object within the universe might be true for the fabric of the universe itself. Namely, just as the tightrope’s surface has both large and small dimensions, so does the fabric of space.
Maybe the three dimensions we all know about - left/right, back/forth, and up/down - are like the horizontal extent of the tightrope, dimensions of the big, easy-to-see variety. But just as the surface of the tightrope has an additional, small, curled-up, circular dimension, maybe the fabric of space also has a small, curled-up, circular dimension, one so small that no one has powerful enough magnifying equipment to reveal its existence. Because of its tiny size, Klein argued, the dimension would be hidden.
With this modification to Kaluza’s original idea, Klein provided an answer to how the universe might have more than the three dimensions of common experience that could remain hidden, a framework that has since become known as Kaluza-Klein theory. And since an extra dimension of space was all Kaluza needed to merge general relativity and electromagnetism, Kaluza-Klein theory would seem to be just what Einstein was looking for.
Indeed Einstein and many others became quite excited about unification through a new, hidden space dimension, and a vigorous effort was launched to see whether this approach would work in complete detail. Einstein continued to dabble in the Kaluza-Klein theory until at least the early 1940s. But the theory encountered difficulties in trying to describe the microworld, and in particular the incorporation of the electron into the extra-dimensional picture.
There was another reason scientists were hesitant about the approach. Many found it both arbitrary and extravagant to postulate a hidden spatial dimension. If you asked Kaluza and Klein why the universe had five spacetime dimensions rather than four, or six, or seven, or 7,000 for that matter, they wouldn’t have had an answer much more convincing than “Why not?”
More than three decades later, the situation changed radically with the advent of string theory,  the first approach to merge general relativity and quantum mechanics, with the potential to unify our understanding of all forces and all matter. But the quantum mechanical equations of string theory don’t work in four spacetime dimensions, nor in five, six or seven, or 7000. Instead, the equations of string theory work only in ten spacetime dimensions - nine of space, plus time. String theory demands more dimensions.
Prior to string theory, no theory said anything at all about the number of spatial dimensions in the universe. Every theory from Newton to Maxwell to Einstein assumed that the universe had three space dimensions, much as we all assume the sun will rise tomorrow. Kaluza and Klein proffered a challenge by suggesting that there were four space dimensions, but this amounted to yet another assumption - a different assumption, but an assumption nonetheless.
Now, for the first time, string theory provided equations that predicted the number of space dimensions. A calculation - not an assumption, not a hypothesis, not an inspired guess - determines the number of space dimensions according to string theory, and the surprising thing is that the calculated number is not three but nine. String theory leads us, inevitably, to a universe with six extra space dimensions and hence provides a compelling, ready-made context for invoking the ideas of Kaluza and Klein. Their original proposal assumed only one hidden dimension, but it’s easily generalized to two, three, or even six extra dimensions required by string theory.
However, there’s an awkward detail regarding string theory. Over the last three decades, not one but five distinct versions of string theory have been developed. While their names are not of the essence, they are called Type 1, Type IIA, Type IIB, Heterotic-O, and Heterotic-E, and they all share the same essential features; the basic ingredients are strands of vibrating energy - and as calculations in the 1970s and 1980s revealed, each theory requires six extra space dimensions; but when they are analyzed in detail, significant differences appear.
During the late 1980s and early 1990s, with many physicists hotly pursuing an understanding of one or another of the string theories, the enigma of the five versions was not a problem researchers typically dealt with on a day-to-day basis. Instead, it was one of those quiet questions that everyone assumed would be addressed in the distant future, when the understanding of each individual string theory had become significantly more refined.
But in the summer of 1995, with little warning, these modest hopes were wildly exceeded when Edward Witten - who for two decades has been the world’s most renowned string theorist - uncovered a hidden unity that tied all five string theories together. Witten showed that rather than being distinct, the five theories are actually just five different ways of mathematically analyzing a single theory. The unifying master theory has tentatively been called M-theory.
Witten’s work revealed that the approximate string theory equations, used in the 1970s and 1980s to conclude that the universe must have nine space dimensions, missed the true number by one. The exact answer, his analysis showed, is that the universe according to M-theory has ten space dimensions, that is eleven spacetime dimensions. 
Much as Kaluza found that a universe with five spacetime dimensions provided a framework for unifying electromagnetism and gravity, and much as string theorists found that a universe with ten spacetime dimensions provided a framework for unifying quantum mechanics and general relativity, Witten found that a universe with eleven spacetime dimensions provided a framework for unifying all string theories.
Following Witten’s paper, the avalanche of subsequent results led to the realization that string theory, and the M-theoretic framework to which it now belongs, contains ingredients besides strings. The analyses showed that there are two-dimensional objects called, naturally enough, membranes or - in deference to systematically naming their higher-dimensional cousins - two-branes.
There are objects with three spatial dimensions called three-branes. And, although increasingly difficult to visualize, the analyses showed that there are also objects with p spatial dimensions, where p can be any whole number less than 10, known - with no derogation intended - as p-branes. Thus strings are but one ingredient in string theory, not the ingredient.
This raises an intriguing possibility. Might we, right now, be living within a three-brane? Like Snow White, whose world exists within a two-dimensional movie screen - a two-brane - that itself resides within a higher-dimensional universe (the three space dimensions of the movie theatre), might everything we know exist within a three- dimensional screen - a three-brane - that itself resides within the higher-dimensional universe of string/M-theory?
Could it be that what Newton, Leibniz, Mach, and Einstein called three-dimensional space is actually a particular three-dimensional entity in string/M-theory? Or, in a more relativistic language, could it be that the four-dimensional spacetime developed by Minkowski and Einstein is actually the wake of a three-brane as it evolves through time? In short might the universe as we know it be a brane? The possibility that we are living within a three-brane - the so-called braneworld scenario - is the latest twist in string/M-theory’s story.
If we are living within a three-brane - if our four-dimensional spacetime is nothing but the history swept out by a three-brane through time - then the venerable question of whether spacetime is a something would be cast in a brilliant new light. Familiar four-dimensional spacetime would arise from a real physical entity in string/M-theory, a three-brane, not from some vague or abstract idea.
In this approach, the reality of our four-dimensional spacetime would be on a par with the reality of an electron or a quark.  But if the universe we’re aware of really is a three-bane, wouldn’t even a casual glance reveal that we are immersed within something - within the three-brane interior?
Well, we’ve already learned of things within which modern physics suggest we may be immersed - a Higgs ocean; space filled with dark energy; myriad quantum field fluctuations - none of which make themselves directly apparent to unaided perceptions. So it shouldn’t be a shock to learn that string/M-theory adds another candidate to the list of invisible things that may fill ‘empty’ space. But let’s not get cavalier.
For each of the previous possibilities, we understand its impact on physics and how we might establish that it truly exists. Indeed, for two of the three - dark energy and quantum fluctuations - we’ve seen that strong evidence supporting their existence has already been gathered; and evidence for the Higgs field is being sought at current and future accelerators. So what is the corresponding situation for life within a three-brane? If the brane-world scenario is correct, why don’t we see the three-brane, and how would we establish that it exists?
The answer highlights how the physical implications of string/M-theory in the braneworld context differ radically from the earlier ‘branefree’ (or, as they’re sometimes affectionately called ‘no-braner’) scenarios. Consider, as an important example, the motion of light - the motion of photons. In string theory, a photon is a particular string vibrational pattern. More specifically, mathematical studies have shown that in the braneworld scenario, only open string vibrations, not closed ones, produce photons, and this makes a big difference.
Open string end-points are constrained to move within the three-brane, but are otherwise completely free. This implies that photons (open strings executing the photon mode of vibration) would travel without any constraint or obstruction throughout our three-brane. And that would make the brane appear completely transparent - completely invisible - thus preventing us from seeing that we are immersed within it.
That’s an intense realization with important consequences. Earlier, we required the extra dimensions of string/M-theory to be tightly curled up. The reason, clearly, is that we don’t see the extra dimensions and so they must be hidden away. And one way to hide them is to make them smaller than we or our equipment can detect.
But let’s now examine the issue in the braneworld scenario. How do we detect things? Well, when we use our eyes, we use the electromagnetic force; when we use powerful instruments like electron microscopes, we also use the electromagnetic force; when we use atom smashers, one of the forces we use to probe the ultrasmall is, once again, the electromagnetic force.
But if the electromagnetic force is confined to our three-brane, our three space dimensions, it is unable to probe the extra dimensions, regardless of their size. Photons cannot escape our dimensions, enter the extra dimensions, and then travel back to our eyes or equipment allowing us to detect the extra dimensions, even if they were as large as the familiar space dimensions.
So, if we live in a three-bane, there is an alternative explanation for why we’re not aware of the extra dimensions. It is not necessarily that the extra dimensions are extremely small. They could be big. We don’t see them because of the way we see. We see by using the electromagnetic force, which is unable to access any dimensions beyond the three we know about. Like an ant walking along a lily pad, completely unaware of the deep waters lying just beneath the visible surface, we could be floating within a grand, expansive, higher-dimensional space, but the electromagnetic force - eternally trapped within our dimensions - would be unable to reveal this.
Okay, you might say. But the electromagnetic force is only one of nature’s four forces. What about the other three? Can they probe into the extra dimensions, thus enabling us to reveal their existence? For the strong and the weak nuclear forces, the answer is, again, no. In the braneworld scenario, calculations show that the messenger particles for these forces - gluons and W and Z particles - also arise from open-string vibrational patterns, so they are just as trapped as photons, and processes involving the strong and weak nuclear forces are just as blind to the extra dimensions.
The same goes for particles of matter. Electrons, quarks, and all the other particle species also arise from the vibrations of open strings with trapped endpoints. Thus, in the braneworld scenario, you and I and everything we’ve ever seen are permanently imprisoned within our three-brane. Taking account of time, everything is trapped within our four-dimensional slice of spacetime.
Well, almost everything. For the force of gravity, the situation is different. Mathematical analyses of the braneworld scenario have shown that gravitons arise from the vibrational pattern of closed strings, much as they do in the previously discussed no-braner scenario. And closed strings - strings with no endpoints - are not trapped by branes. They are as free to leave a brane as they are to roam on through it.
So, if we were living in a brane, we would not be completely cut off from the extra dimensions. Through the gravitational force, we could both influence and be influenced by the extra dimensions. Gravity, in such a scenario, would provide our sole means for interacting beyond our three space dimensions.
How big could the extra dimensions be before we’d become aware of them through the gravitational force? Hundreds of years of experiments have confirmed that gravity varies inversely with the square of distance, giving strong evidence that there are three space dimensions. But as of 1998, no experiment had ever probed gravity’s strength on separations smaller than a millimetre.
This led to the proposal that in the braneworld scenario extra dimensions could be as large as a millimetre and would still have been undetected. This radical suggestion inspired a number of experimental groups to initiate a study of gravity at submillimeter distance in hopes of finding violations of the inverse square law; so far, none have been found, down to a tenth of a millimetre. Thus, even with today’s state-or-the-art gravity experiments, if we are living within a three-brane, the extra dimensions could be as large as a tenth of a millimetre, and yet we wouldn’t know it.
This is one of the most striking realizations of the last decade. Using the three nongravitational forces, we can probe down to about a billionth of a billionth (10-18) of a metre, and no one has found any evidence of extra dimensions. 
But in the braneworld scenario, the nongravitational forces are impotent in searching for extra dimensions since they are trapped on the brane itself. Only gravity can give insight into the nature of the extra dimensions, and, as of today, the extra dimensions could be as thick as a human hair and yet they’d be completely invisible to our most sophisticated instruments.
Right now, right next to you, right next to me, and right next to everyone else, there could be another spatial dimension - a dimension beyond left/right, back/forth, and up/down, a dimension that’s curled up but still large enough to swallow something as thick as this page - that remains beyond our grasp 
Over the last century,  we’ve become intimately acquainted with some previously hidden features of space and time through Einstein’s two theories of relativity and through quantum mechanics. The slowing of time, the relativity of simultaneity, alternative slicings of spacetime, gravity as the warping and curving of space and time, the probabilistic nature of reality, and long-range quantum entanglement were not on the list of things that even the best of the world’s nineteenth century physicists would have expected to find just around the corner. And yet there they were, as attested to by both experimental results and theoretical explanations.
In our age, we’ve come upon our own panoply of unexpected ideas. Dark matter and dark energy that appear to be, far and away, the dominant constituents of the universe. Gravitational waves - ripples in the fabric of spacetime - which were predicted by Einstein’s relativity and may one day allow us to peek further back in time than ever before. A Higgs ocean, which permeates all of space and which, if confirmed, will help us to understand how particles acquire mass. Inflationary expansion, which may explain the shape of the cosmos, resolve the puzzle of why it’s so uniform on large scales, and set the direction to time’s arrow.
String theory, which posits loops and snippets of energy in place of point particles and promises a bold version of Einstein’s dream in which all particles and all forces are combined into a single theory. Extra space dimensions emerging from the mathematics of string theory, and possibly detectable in accelerator experiments during the next decade. A braneworld, in which our three space dimensions may be but one universe among many, floating in a higher-dimensional spacetime. And perhaps even emergent spacetime, in which the very fabric of space and time is composed of more fundamental spaceless and timeless entities.
During the next decade, even more powerful accelerators will provide much needed experimental input, and many physicists are confident that data gathered from the highly energetic collisions that are planned will confirm a number of these pivotal theoretical constructs. I share this enthusiasm and eagerly await the results.
Until our theories make contact with observable, testable phenomena, they remain in limbo - they remain promising collections of ideas that may or may not have relevance for the real world. The new accelerators will advance the overlap between theory and experiment substantially, and, we physicists hope, will usher many of these ideas into the realm of established science.
 Source: Chapters 12 & 13 in The Fabric of the Cosmos by Brian Greene. The published text, from page 360 to 400 in the Penguin 2007 edition, has been compressed by four-fifths to give a flavour of how our everyday ‘three space and one time dimension’ reality is being extended at the leading edge of theoretical physics as new facts about this reality emerge. [Ed]
 A string is a one-dimensional vibrating filament of energy, superseding atoms and particles as the smallest unit from which protons, quarks etc. are constituted. Superstring theory incorporates these vibrating strings as one dimensional loops (closed strings) or snippets (open strings) to unite general relativity, quantum mechanics and supersymmetry. [Ed]
 Lethbridge was very specific that ‘length’ has a rate of 13⅓-inches and ‘thickness’ a rate of 26⅔. A further 13⅓ would give a rate of 40-inches. Might this be the third space dimension? Lethbridge would have been aware of the significance of this possibility (hence the precision of the thirds), but I am not aware of any mention of this in his published books. Three whorls would be required to give the nine space dimensions of string theory, while a fourth would be needed to accommodate the needs of string/M-theory’s ten space dimensions. Perhaps somebody in London’s Docklands might care to experiment with left/right, back/forth and in/out in one of the atriums in the City of London’s Financial District?[Ed]
 You could still ask whether the larger spacetime within which strings and branes exist - the eleven dimensions of string/M-theory - is itself an entity; the reality of the spacetime arena we directly experience, though, would be rendered obvious.
 The strings (of string theory) are so small that a direct observation would be tantamount to reading the text on this page from a distance of 100 light years: it would require resolving power nearly a billion billion times finer than our current technology allows.’ Brian Greene in The Fabric of the Cosmos (page 352). [Ed]
 There is even a suggestion that gravity itself can be trapped, not by a sticky brane, but by extra dimensions that curve in just the right way, relaxing even further the constraints on their size.
 The last five paragraphs have been taken from the final chapter of The Fabric of the Cosmos (page 492 in the Penguin 2007 edition). [Ed]