Most of us have gotten used to the idea of there being 4 dimensions: but how can we possibly imagine the 10th? The video series you'll see inside this article starts from the unique argument that time really is just one of the directions in the 4th spatial dimension, and our spacetime universe is being created one planck length at a time as we twist and turn in the available branches of the 5th dimension.

For many people, this "new way of thinking about time and space" has resonances with their own ways of understanding reality.

For many people, this "new way of thinking about time and space" has resonances with their own ways of understanding reality.

The following is excerpted from Active Consciousness: Awakening the Power Within, published by R.L.Ranch Press.

*At each moment of time— -- the Now -- you stand at the gateway to an infinite number of possible futures in higher dimensional space. This gateway is the Choice Point.*

Some of these futures are more likely to occur than others. If you do nothing, one of these probable futures will play itself out in a fairly predictable, even mechanistic way.

Some of these futures are more likely to occur than others. If you do nothing, one of these probable futures will play itself out in a fairly predictable, even mechanistic way.

One of the ways you can exert a force upon this unfolding is to use explicit action. Another way is through the subtle force of Active Consciousness, which allows you to enable unlikely unfoldings to occur.

According to the Columbia Encyclopedia, a field is a “region throughout which a force may be exerted.” Back in the 1800s, chemist and physicist Michael Faraday developed this concept in order to explain the mysterious properties of magnets. People at the time were perplexed by the way magnets could exert instantaneous forces upon metal objects when nothing seemed to be transmitted between them. Faraday hypothesized that magnets are

…Fields can also be related to one another. For example, the magnetic field is related to the electric field because electricity can be used to create a magnet and a magnet can be used to create electricity. For this reason, the two fields are actually considered to be aspects of the same field—the electromagnetic field. … [One] important question in modern physics is whether all fields are, in fact, interrelated and aspects of a single underlying field—a

So how does consciousness fit into all of this? If you think about it,

Things might go in the other direction as well. Not only may we feel the effects of the consciousness field, but we might also be able to exert our own forces upon it. In other words,

Of course, from the standpoint of quantum physics, much of this is old hat. It is already accepted by physicists that, at the subatomic level, everything is a set of probabilities or potentialities affected by awareness. Even a simple act of observation—or, indeed, an

Perhaps we are wrong. …Experiments with random event generators… support the possibility that our thoughts and intentions can cause observable changes in the world. Another set of experiments, conducted by researchers on the effects of transcendental meditation, found that if enough meditators work collectively, the amount of violent crime in a city can be lowered for extended periods of time. The truth may be that our beliefs and intentions

… So if we

Let me begin by saying that when I say “higher dimensions,” I’m not talking about figurative realms populated by mysterious beings. I’m literally talking about higher spatial dimensions. Just as a two-dimensional universe can be understood as a flat surface, and just as we ourselves live in three-dimensional space with objects possessing length, width, and height, a four-dimensional universe would actually be a space with one extra spatial dimension. While it’s easy to visualize other ways of adding a “dimension” to our reality—for example, a dimension of time [it]would add a timestamp to each object, and a dimension of *color* would add a color—getting our brains to wrap around an extra spatial dimension is quite difficult.

…It turns out that a fascination with the fourth dimension was a fad of the late 1800s. This was the era in which mediums, like today’s television medium John Edward, were quite popular. It was also the era in which*spiritualism* emerged, an American religious movement focused on communicating with the spirits of those who had passed away. Scientists of the time proposed the fourth spatial dimension as a way of explaining spiritist phenomena. Ultimately, however, nearly all the mediums of the 1800s were debunked as frauds and as a consequence, speculation about the fourth dimension fell into disrepute as well. But perhaps we should not have been so hasty to throw the baby out with the bath water.

As mathematician Rudy Rucker points out in his book,*The Fourth Dimension* [9], the first philosopher to discuss the possibility of a fourth dimension was Immanuel Kant (1724-1804). The idea of spirits as four-dimensional beings was then popularized by Johann Carl Friederich Zollner (1834-1882), an astronomer at the University of Leipzig. The first true theoretician of the fourth dimension, however, was British mathematician Charles Hinton (1853-1907), known for his work on visualizing the geometry of higher dimensional space. In addition to his mathematical work, Hinton also wrote a series of science fiction books, such as *What is the Fourth Dimension?* and *A Plane World*, which focus on realms with different dimensionality than our own. He also believed that higher spatial dimensions could be used to prove the inherently unified nature of the universe.

Hinton came up with a variety of ways to help us understand the fourth dimension and its properties. One was… [examining] the difference between two and three dimensions and then, by analogy, grasping the difference between three and four dimensions. Here’s [an] illustration of this technique.

If you take two flat two-dimensional surfaces and intersect them in the context of three-dimensional space, what do you get? The answer: a line, which is a one-dimensional object. To understand this, just imagine taking two sheets of paper and intersecting them. By analogy, if you take two three-dimensional spaces and intersect them in the context of four-dimensional space, what will you get? The answer: a two-dimensional space—a flat surface. That means, if there were another three-dimensional world floating out there and it happened to overlap and intersect with us, the region of overlap between our two worlds would appear to us like a flat, impossibly thin sheet of paper. Could this be an explanation for the phenomenon of ghosts? Are ghosts simply ephemeral two-dimensional visions of beings in another three-dimensional world? That’s what Zollner proposed in the mid 1800s…

…It turns out that a fascination with the fourth dimension was a fad of the late 1800s. This was the era in which mediums, like today’s television medium John Edward, were quite popular. It was also the era in which

As mathematician Rudy Rucker points out in his book,

Hinton came up with a variety of ways to help us understand the fourth dimension and its properties. One was… [examining] the difference between two and three dimensions and then, by analogy, grasping the difference between three and four dimensions. Here’s [an] illustration of this technique.

If you take two flat two-dimensional surfaces and intersect them in the context of three-dimensional space, what do you get? The answer: a line, which is a one-dimensional object. To understand this, just imagine taking two sheets of paper and intersecting them. By analogy, if you take two three-dimensional spaces and intersect them in the context of four-dimensional space, what will you get? The answer: a two-dimensional space—a flat surface. That means, if there were another three-dimensional world floating out there and it happened to overlap and intersect with us, the region of overlap between our two worlds would appear to us like a flat, impossibly thin sheet of paper. Could this be an explanation for the phenomenon of ghosts? Are ghosts simply ephemeral two-dimensional visions of beings in another three-dimensional world? That’s what Zollner proposed in the mid 1800s…

Of course, speculation about higher dimensions is nothing new to physicists; they have considered the possible existence of four or more spatial dimensions for a long time. For example, between 1907 and 1915, Einstein developed the theory of *general relativity*, which states that gravity exists because a large mass causes three-dimensional space to bend within the fourth dimension. This is easy to visualize if you drop down into two dimensions. What if the mass of a large Circle caused the flat world of two dimensions to bend around it in three-dimensional space? Just imagine bending a large piece of paper around the edges of the Circle. That would cause everything near the Circle to fall toward it. In the same way, bending three-dimensional space around massive bodies like planets would cause nearby objects to fall toward them. Voila! Gravity.

And what about wormholes? …Physicists believe that three-dimensional space might be bent so heavily by the mass of a black hole that the fabric of the universe folds back on itself, causing one part to intersect with another, thereby creating a portal between them. To see how this could happen, just imagine bending a sheet of paper so much that two distant points on it touch.

…One question that still begs to be answered, though, is: What kind of access can*we* have to the fourth or higher dimensions? Even if our bodies are stuck here in three-dimensional space, is there a four-dimensional aspect of ourselves that we’re simply not aware of? Ouspensky, Gurdjieff’s student and chronicler, thought so. In fact, he felt that the primary goal of Gurdjieff’s teachings was to help us access our higher dimensional selves. As Rudy Rucker writes, “For Ouspensky, the fourth dimension was not only a spatial concept but a type of consciousness, an awareness of greater complexities and higher unities.” [10] Thus, rather than trying to contact other beings in the fourth dimension, perhaps our real goal should be to tap into our own four dimensionality. Indeed, I believe it is from this higher dimensional perspective that we may be able to exert the deep power of active consciousness—the ability to navigate and influence the unfolding of our three-dimensional lives.

And what about wormholes? …Physicists believe that three-dimensional space might be bent so heavily by the mass of a black hole that the fabric of the universe folds back on itself, causing one part to intersect with another, thereby creating a portal between them. To see how this could happen, just imagine bending a sheet of paper so much that two distant points on it touch.

…One question that still begs to be answered, though, is: What kind of access can

… [It turns out that] from the perspective of the fourth dimension, …a complete human life—every activity of the inner and outer physical body—would simply appear as an eternal shape or object in four-dimensional space. …[But] what determines [its] shape? That’s when things get interesting. At each point in time we make choices. Do we walk left or right? Does our body repair itself or does it descend further into disease? At each instant, we choose one from a potentially infinite number of possible futures that lie before us. And while it may seem that only one future is chosen at each point in time, perhaps all of the other possible choices and futures exist in four-dimensional space too. If so, then our many *potential* lives—from a four-dimensional perspective—would look like a vast branching tree of possibilities.

Interestingly, this idea was hypothesized by physicist Hugh Everett in 1957. In the quantum realm, particles exist in many simultaneous states until they are observed and become “solidified” into a single state. To us, these quantum choices seem random. Everett proposed, however, that there really is no randomness at all. Each and every one of the possible choices for a particle actually exists—*in another parallel world*. We ourselves perceive only one choice—the one that takes place in our world. But Everett asserted that the others choices exist too—in other worlds. Indeed, parallel versions of ourselves exist in those other worlds too, and they witness the particle resolving into other possible states. Mind boggling! It sounds like science fiction, and indeed, many science fiction stories have been written based on this idea. But what if it’s science fact? I believe that this conception of our universe may also be the basis for how active consciousness might operate.

Interestingly, this idea was hypothesized by physicist Hugh Everett in 1957. In the quantum realm, particles exist in many simultaneous states until they are observed and become “solidified” into a single state. To us, these quantum choices seem random. Everett proposed, however, that there really is no randomness at all. Each and every one of the possible choices for a particle actually exists—

Let’s begin to consider this mind-bending idea by simply imagining a single point of branching. Consider the world of a three-dimensional ball, depicted in Figure 1. The ball begins, at time *t1*, at the bottom of a road. When it arrives at point *C* (which occurs at time*t2*), the ball must make a choice. Does it move along the branch moving to the right, or does it continue moving upward? In one future reality, the ball has moved up at time *t3*. In another future reality, it has branched to the right at *t3*, creating a fork in four-dimensional space.

Let’s call*C* a *choice point*—a point in space and time at which the Now splits into two or more possible futures. In order for this split to occur, some force must be exerted. If this does not happen, there may be only one possible future—the one that would play itself out according to the mechanistic laws of nature. For the ball, this might be the future in which it continues moving upward, in a straight line. But if some kind of force, intention, or will is brought to bear at point C, the Now will split into more than one possible future, like a branching tree. And when we humans exert such a force upon our own lives, we experience the sensation of *free will*.

Of course, people usually make rather mundane choices at choice points. We might choose to move our arm or walk to the grocery store. But at times the actions we take can be much more subtle—like when our thoughts affect the behavior of a random event generator.*What I would like to propose is that this more subtle kind of force is what underlies the power of active consciousness.* I’ll call it the *C-force*. And just as the experiments with random event generators showed, a person’s use of the C-force can influence not only his or her own life, but can also affect the unfolding of the greater reality around us. As a result, each and every one of us helps to create a much greater collective reality—an infinitely complex shape in four-dimensional space. Indeed, even if we took into account every tiny mechanism already understood by conventional science, the influence of our collective use of the C-force would be beyond our imaginations! In fact, it might even be possible that our influence extends *beyond* four dimensions. If so, we humans—seemingly three-dimensional creatures—may have creative potential that we have only just begun to tap into.

Let’s call

Of course, people usually make rather mundane choices at choice points. We might choose to move our arm or walk to the grocery store. But at times the actions we take can be much more subtle—like when our thoughts affect the behavior of a random event generator.

…When I chose the term C-force, I used the letter “C” intentionally—because it serves as a reminder of several important aspects of this force: Consciousness, Choice, and Creativity. Using the C-force—the force of active consciousness—you can make beneficial choices that take you down desired paths in life. You can create new and unlikely paths as well. Either way, harnessing the power of active consciousness will enable you to make the improbable much more probable and become an active creator of your own destiny. Let’s examine this process now in more detail.

…One way in which the C-force might be used is through the process of*manifestation*. In this case, you enable an unlikely combination of otherwise mundane events to occur so that a desired goal comes about. For example, let’s say that you would like to get a desirable parking spot near a restaurant. The restaurant might be in the middle of a busy city where parking spots are hard to find, but finding the perfect spot at just the right time could definitely happen. Everything just needs to be coordinated correctly: the choices made by the person who parked in the spot before you, the route you choose to drive, the timing of the lights as you are driving, and so on. By invoking the power of active consciousness… , these fortuitous choices and events could be enabled.

Or let’s say that you would like to find a new job with better pay. You’ve been stuck in a rut for a long time and can’t figure out how to leave your current job situation. Through the process of manifestation, however, it may be quite possible that a sequence of events could occur that leads you to your goal. Perhaps a bout with the flu forces you to take a couple of weeks off from work. As a result, your quirky boss decides to lay you off. Now you’ve been forced to leave your job. The following week, a long-lost friend calls you unexpectedly. She tells you about an acquaintance who needs to hire someone with your exact qualifications—the dream job. With this connection, you get the new job with ease.

Another way the C-force might operate is through the process of creation. In this case, a bit more magic is involved because much more unlikely (but still possible) unfoldings occur. Whereas manifestation is about enabling an unlikely*combination* of events to occur, creation is more about enabling more unlikely choices and events to appear in the first place.

For example, let’s say that you suffer from chronic eczema. Given the natural tendency of the body to recreate itself in the same way, it is most likely that your skin will continue to suffer from this condition. You might apply some cortisone cream, which chemically forces the body to suppress the eczema, but the innate tendency for your skin to develop eczema has not gone away. The next time you experience a period of anxiety or come into contact with an allergen, the eczema flares up.

But remember this: your skin is always sloughing off and regenerating. The outer layer of your skin (the epidermis) is replaced every month. There is a*possibility* that your skin could regenerate without this problem and that it would never return again. It’s not very probable, but it is possible. Similarly, it is *possible*that a cancerous tumor could be broken down by the natural defenses of the body without the use of poisonous chemotherapy or toxic radiation. It’s not very probable, but it *is* possible.

I believe that the C-force—the force of active consciousness—can be used to create such improbable but possible choices for the body. For example, a patient could use the power of active consciousness to enable their skin to regenerate without eczema or their cancerous tumor to be broken down and absorbed. Such medical cases of “spontaneous remission” do occur. Because doctors cannot understand or explain them, they sweep them under the rug by saying that their original diagnosis was mistaken or that the cure was an unexplainable fluke. But what if more of us could tap into this kind of healing through the C-force—the*Cure* force? It would be wonderful! We could be spared many toxic medicines and expensive medical bills too.

In fact, “energy” medicines like homeopathy, acupuncture, and hands-on healing may make this particular application of the C-force substantially easier to achieve. That’s because they operate not only on the physical body, but also on a subtler aspect of our selves—what is often called the “energy body”…. By helping to dislodge problems within this invisible realm—a realm that most alternative medical systems view as the true origin of disease—such treatments vastly increase the*probability* that the physical body will be able to create a healthier future. As a result, they also make healing applications of the C-force easier to achieve.

…One way in which the C-force might be used is through the process of

Or let’s say that you would like to find a new job with better pay. You’ve been stuck in a rut for a long time and can’t figure out how to leave your current job situation. Through the process of manifestation, however, it may be quite possible that a sequence of events could occur that leads you to your goal. Perhaps a bout with the flu forces you to take a couple of weeks off from work. As a result, your quirky boss decides to lay you off. Now you’ve been forced to leave your job. The following week, a long-lost friend calls you unexpectedly. She tells you about an acquaintance who needs to hire someone with your exact qualifications—the dream job. With this connection, you get the new job with ease.

Another way the C-force might operate is through the process of creation. In this case, a bit more magic is involved because much more unlikely (but still possible) unfoldings occur. Whereas manifestation is about enabling an unlikely

For example, let’s say that you suffer from chronic eczema. Given the natural tendency of the body to recreate itself in the same way, it is most likely that your skin will continue to suffer from this condition. You might apply some cortisone cream, which chemically forces the body to suppress the eczema, but the innate tendency for your skin to develop eczema has not gone away. The next time you experience a period of anxiety or come into contact with an allergen, the eczema flares up.

But remember this: your skin is always sloughing off and regenerating. The outer layer of your skin (the epidermis) is replaced every month. There is a

I believe that the C-force—the force of active consciousness—can be used to create such improbable but possible choices for the body. For example, a patient could use the power of active consciousness to enable their skin to regenerate without eczema or their cancerous tumor to be broken down and absorbed. Such medical cases of “spontaneous remission” do occur. Because doctors cannot understand or explain them, they sweep them under the rug by saying that their original diagnosis was mistaken or that the cure was an unexplainable fluke. But what if more of us could tap into this kind of healing through the C-force—the

In fact, “energy” medicines like homeopathy, acupuncture, and hands-on healing may make this particular application of the C-force substantially easier to achieve. That’s because they operate not only on the physical body, but also on a subtler aspect of our selves—what is often called the “energy body”…. By helping to dislodge problems within this invisible realm—a realm that most alternative medical systems view as the true origin of disease—such treatments vastly increase the

…So how does this fit within the four-dimensional model described earlier? … In Figure 2, I use circles to represent choice points and arrows to indicate possible futures that emanate from them. Let’s say that you are currently at NOW. The circle labeled GOAL is a future that you’d like to reach—say, one in which you have a new job. The circle labeled MOST LIKELY FUTURE is the most likely or probable outcome—the one in which you stay at your current job. As the diagram illustrates, you can reach your goal in at least two ways—either by choosing or creating a new path right now (branching upward immediately), or by doing so a little later on. For instance, right NOW, you might decide to quit your job. Or you might create an improbable future—through creation—where a freak accident or illness ultimately leads to you being laid off. By using active consciousness to enable new choices to appear and to help you make the correct choices over time, you may find that you are ultimately led to your goal of a new and better job.

…In many ways, it’s all about possibilities and probabilities. Even if something is improbable, it can still be possible. And if it’s possible, the force of active consciousness—the C-force—can play a part in making it happen. Using manifestation, you can make the right choices at the right time and, through an unlikely combination of such choices, you are led to your goal. Using creation, an unlikely choice [may unexpectedly appear] before you; you just need to take it.

]]>…In many ways, it’s all about possibilities and probabilities. Even if something is improbable, it can still be possible. And if it’s possible, the force of active consciousness—the C-force—can play a part in making it happen. Using manifestation, you can make the right choices at the right time and, through an unlikely combination of such choices, you are led to your goal. Using creation, an unlikely choice [may unexpectedly appear] before you; you just need to take it.

The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move.

For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true, as some have suggested, that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?

For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true, as some have suggested, that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?

According to the early 20th century horror writer H.P. Lovecraft, these higher dimensions do indeed exist, and are home to all manner of evil creatures. In Lovecraft's mythology, the most terrible of these beings goes by the name of Yog-Sothoth. Interestingly, on the rare occasions that Yog-Sothoth appears in the human realm, it takes the form of "a congeries of iridescent globes... stupendous in its malign suggestiveness".

Lovecraft had some interest in mathematics, and indeed used ideas such as hyperbolic geometry to lend extra strangeness to his stories (as Thomas Hull has discussed in Math Horizons). But he could not have known how fortunate was the decision to represent Yog-Sothoth in this manner. Strange spheres really are the keys to higher dimensional worlds, and our understanding of them has increased greatly in recent years. Over the last 50 years a subject called*differential topology* has grown up, and revealed just how alien these places are.

Lovecraft had some interest in mathematics, and indeed used ideas such as hyperbolic geometry to lend extra strangeness to his stories (as Thomas Hull has discussed in Math Horizons). But he could not have known how fortunate was the decision to represent Yog-Sothoth in this manner. Strange spheres really are the keys to higher dimensional worlds, and our understanding of them has increased greatly in recent years. Over the last 50 years a subject called

Do higher dimensions exist? Mathematics provides a surprisingly emphatic answer to this question. Just as a 2-dimensional plane can be described by pairs of coordinates such as (5,6) with reference to a pair of axes, so 3-dimensional space can be described by triples of numbers such as (5,6,3). Of course we can continue this line of thought: 4-dimensional space, for a mathematician, is identified with the sets of quadruples of real numbers, such as (5,6,3,2). This procedure extends to all higher dimensions. Of course this does not answer the physicist's question, of whether such dimensions have any objective physical existence. But mathematically, at least, as long as you believe in numbers, you don't have much choice but to believe in 4-dimensional space too.

Well that is fine, but how can such spaces be imagined? What does the lair of Yog-Sothoth actually look like? This is a much harder question to answer, since our brains are not wired to see in more dimensions than three. But again, mathematical techniques can help, firstly by allowing us to generalise the phenomena that we do see in more familiar spaces.

An important example is the sphere. If you choose a spot on the ground, and then mark all the points which are exactly 1cm away from it, the shape that emerges is a circle, with radius 1cm. If you do the same thing, but in 3-dimensional space, we get an ordinary sphere or globe. Now comes the exciting part, because exactly the same trick works in four dimensions, and produces the first*hypersphere*.

What does this look like? Well, when we look at the circle from close up, each section looks like an ordinary 1-dimensional line (so the circle is also known as the 1-sphere). The difference between the circle and the line is that when viewed from afar, the whole thing curves back to connect to itself, and has only finite length. In the same way, each patch of the usual sphere (that is to say, the 2-sphere) looks like a patch of the 2-dimensional plane. Again, these patches are sewn together in a way that leaves no edges, and has only finite area. So far, so predictable, but exactly the same thing is true for the first hypersphere (or 3-sphere): each region looks just like familiar 3-dimensional space. We might be living in one now, for all we can see. But just like its lower dimensional cousins, the whole thing curves around on itself, in a way that flat 3-dimensional space does not, producing a shape with no sides, and only finite volume. Of course we do not stop here: the next hypersphere (the 4-sphere), is such that every region looks like 4-dimensional space, and so on in every dimension.

Well that is fine, but how can such spaces be imagined? What does the lair of Yog-Sothoth actually look like? This is a much harder question to answer, since our brains are not wired to see in more dimensions than three. But again, mathematical techniques can help, firstly by allowing us to generalise the phenomena that we do see in more familiar spaces.

An important example is the sphere. If you choose a spot on the ground, and then mark all the points which are exactly 1cm away from it, the shape that emerges is a circle, with radius 1cm. If you do the same thing, but in 3-dimensional space, we get an ordinary sphere or globe. Now comes the exciting part, because exactly the same trick works in four dimensions, and produces the first

What does this look like? Well, when we look at the circle from close up, each section looks like an ordinary 1-dimensional line (so the circle is also known as the 1-sphere). The difference between the circle and the line is that when viewed from afar, the whole thing curves back to connect to itself, and has only finite length. In the same way, each patch of the usual sphere (that is to say, the 2-sphere) looks like a patch of the 2-dimensional plane. Again, these patches are sewn together in a way that leaves no edges, and has only finite area. So far, so predictable, but exactly the same thing is true for the first hypersphere (or 3-sphere): each region looks just like familiar 3-dimensional space. We might be living in one now, for all we can see. But just like its lower dimensional cousins, the whole thing curves around on itself, in a way that flat 3-dimensional space does not, producing a shape with no sides, and only finite volume. Of course we do not stop here: the next hypersphere (the 4-sphere), is such that every region looks like 4-dimensional space, and so on in every dimension.

Like geometry, topology is a branch of mathematics which studies shapes. One of the fundamental questions to ask is when two shapes are really the same. This does not have a unique answer, it depends on the aspects of the shape that you are most interested in. At a basic level, if two shapes are identical, but are situated in different places, then for most purposes we will count them as being "the same".

Topology has a much broader notion of sameness than geometry. Here, two shapes are deemed "the same" if one can be pulled, stretched and twisted into the form of the other. So, to a topologist, triangles, trapeziums, septagons, and so on are all the same: they are all just circles. On the other hand, a figure of 8 is a genuinely different shape, because the topological definition of sameness never extends to cutting or gluing the shape. So an 8 can never be pulled into the shape of a circle as cutting is forbidden, and neither can a lower case i, as the two parts cannot be glued together.

If you are interested in things like angles, lengths, or areas, then the topological viewpoint is the wrong one. But a lot of important data is retained at this level: a famous example is the London tube map. Here, it is not the lengths or precise routes of the tunnels which matter, but things like the orders of the stations, and the ways that different tube lines intersect. These phenomena are topological in nature, and survive topological morphing. This is convenient, as it allows Londoners to use the famous simplified schematic map, rather than a detailed map of the whole city, incorporating the exact routes of all the tube lines.

Some shapes, such as the donut-shaped torus, have holes in them. These holes are essential; they cannot be removed by topological twisting or stretching. But which are the shapes with no holes? The most famous theorem in topology, the*Poincaré conjecture*, provides an elegant answer to this question: it says that the only such shapes are the spheres. This is not true from a geometrical viewpoint, as cubes, pyramids, dodecahedra, and a multidue of other shapes all have no holes. But, of course, to a topologist, all these exciting shapes are nothing more than spheres.

We have known since 2002 that the Poincaré conjecture is indeed true. Henri Poincaré's original question concerned the 3-sphere, but in fact exactly the same thing applies in all higher dimensions too. The fact is that, when viewed topologically, spheres are beautifully simple and unique objects in every dimension. However, in 1956 the first evidence arrived that a slight change in perspective would make the story hugely more complicated. When approached through the new subject of differential topology, higher dimensional spaces began to reveal some of their extraordinary secrets.

If you are interested in things like angles, lengths, or areas, then the topological viewpoint is the wrong one. But a lot of important data is retained at this level: a famous example is the London tube map. Here, it is not the lengths or precise routes of the tunnels which matter, but things like the orders of the stations, and the ways that different tube lines intersect. These phenomena are topological in nature, and survive topological morphing. This is convenient, as it allows Londoners to use the famous simplified schematic map, rather than a detailed map of the whole city, incorporating the exact routes of all the tube lines.

Some shapes, such as the donut-shaped torus, have holes in them. These holes are essential; they cannot be removed by topological twisting or stretching. But which are the shapes with no holes? The most famous theorem in topology, the

We have known since 2002 that the Poincaré conjecture is indeed true. Henri Poincaré's original question concerned the 3-sphere, but in fact exactly the same thing applies in all higher dimensions too. The fact is that, when viewed topologically, spheres are beautifully simple and unique objects in every dimension. However, in 1956 the first evidence arrived that a slight change in perspective would make the story hugely more complicated. When approached through the new subject of differential topology, higher dimensional spaces began to reveal some of their extraordinary secrets.

The difference between plain topology and differential topology seems very subtle, but turns out to have astonishing consequences. It hinges on the precise type of pulling and stretching which are allowed during the morphing process. This has a dramatic impact on the shapes which are deemed to be "the same".

The divide is between processes which are*continuous*, meaning that they do not jump or tear, and others which are *smooth*. Smoothness is a much stronger condition than mere continuity. The same distinction applies to shapes themselves: circles and spheres are examples of smooth shapes, while squares and cubes are not smooth because of their sharp edges and corners. All of these are continuous, however, because their edges do not have any gaps or jumps. (A discontinuous line is one which comes in two separate pieces.) There are even fractal patterns which are continuous everywhere, but not smooth anywhere.

In the same way, we can distinguish between morphing which is truly smooth, and that which is merely continuous, but potentially very jerky and violent. It is not at all obvious, however, that this distinction should really matter very much. Might it really be possible that two shapes (also called*manifolds* by topologists) could be the same from a topological perspective (in technical terms, be *homeomorphic*), but not the same from a differential perspective (they are not *diffeomorphic*)? In other words, can we have two shapes that can be morphed into each other without cutting, but for which the morphing can't be smooth, it requires jerks and jumps? This is certainly difficult to imagine, not least because it never happens in dimensions 1, 2, or 3.

The divide is between processes which are

In the same way, we can distinguish between morphing which is truly smooth, and that which is merely continuous, but potentially very jerky and violent. It is not at all obvious, however, that this distinction should really matter very much. Might it really be possible that two shapes (also called

In 1956, John Milnor was investigating 7-dimensional manifolds when he found a shape which seemed very strange. On one hand, it contained no holes, and so it seemed to be a sphere. On the other hand, the way it was curved around was not like a sphere at all. Initially Milnor thought that he had found a counterexample to the 7-dimensional version of the Poincaré conjecture: a shape with no holes, which was not a sphere. But on closer inspection, his new shape could morph into a sphere (as Poincaré insists it must be able to do), but - remarkably - it could not do so smoothly. So, although it was topologically a sphere, in differential terms it was not.

Milnor had found the first*exotic sphere*, and he went on to find several more in other dimensions. In each case, the result was topologically spherical, but not differentially so. Another way to say the same thing is that the exotic spheres represent ways to impose unusual notions of distance and curvature on the ordinary sphere.

In dimensions one, two, and three, there are no exotic spheres, just the usual ones. This is because the topological and differential viewpoints do not diverge in these familiar spaces. Similarly in dimensions five and six there are only the ordinary spheres, but in dimension seven, suddenly there are 28. In higher dimensions the number flickers around between 1 and arbitrarily large numbers:

Milnor had found the first

In dimensions one, two, and three, there are no exotic spheres, just the usual ones. This is because the topological and differential viewpoints do not diverge in these familiar spaces. Similarly in dimensions five and six there are only the ordinary spheres, but in dimension seven, suddenly there are 28. In higher dimensions the number flickers around between 1 and arbitrarily large numbers:

Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

Number of spheres | 1 | 1 | 1 | ? | 1 | 1 | 28 | 28 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 |

The realm which remains the most mysterious, even today, is 4-dimensional space. No exotic spheres have yet been found here. At the same time no-one has managed to prove that none can exist. The assertion that there are no exotic spheres in four dimensions is known as the *smooth Poincaré* conjecture. In case anyone has got this far and is still not sure, let me make this clear: the smooth Poincaré conjecture is not the same thing as the Poincaré conjecture! Among other differences, the Poincaré conjecture has been proved, but the smooth Poincaré conjecture remains stubbornly open today.

So, is the smooth Poincaré conjecture true? Most mathematicians lean towards the view that it is probably false, and that 4-dimensional exotic spheres are likely to exist. The reason is that 4-dimensional space is already known to be a very weird place, where all sorts of surprising things happen. A prime example is the discovery in 1983 of a completely new type of shape in 4-dimensions, one which is completely unsmoothable.

As discussed above, a square is not a smooth shape because of its sharp corners. But it can be smoothed. That is to say, it is topologically identical to a shape which is smooth, namely the circle. In 1983, however, Simon Donaldson discovered a new class of 4-dimensional manifolds which are unsmoothable: they are so full of essential kinks and sharp edges that there is no way of ironing them all out.

Beyond this, it is not only spheres which come in exotic versions. It is now known that 4-dimensional space itself (or R4) comes in a variety of flavours. There is the usual flat space, but alongside it are the exotic R4s. Each of these is topologically identical to ordinary space, but not differentially so. Amazingly, as Clifford Taubes showed in 1987, there are actually infinitely many of these alternative realities. In this respect, the fourth dimension really is an infinitely stranger place than every other domain: for all other dimensions*n*, there is only ever one version of Rn. Perhaps after all, the fourth dimension is the right mathematical setting for the weird worlds of science fiction writers' imaginations.

]]>As discussed above, a square is not a smooth shape because of its sharp corners. But it can be smoothed. That is to say, it is topologically identical to a shape which is smooth, namely the circle. In 1983, however, Simon Donaldson discovered a new class of 4-dimensional manifolds which are unsmoothable: they are so full of essential kinks and sharp edges that there is no way of ironing them all out.

Beyond this, it is not only spheres which come in exotic versions. It is now known that 4-dimensional space itself (or R4) comes in a variety of flavours. There is the usual flat space, but alongside it are the exotic R4s. Each of these is topologically identical to ordinary space, but not differentially so. Amazingly, as Clifford Taubes showed in 1987, there are actually infinitely many of these alternative realities. In this respect, the fourth dimension really is an infinitely stranger place than every other domain: for all other dimensions