Chapter 3

Section 3: Extra Dimensions

Our task now is to dis­cover those addi­tional dimen­sions — to extend the reach of our imag­i­na­tion so that a four-dimensional map no longer restricts our intu­ition. At that point, we may just find that the mys­teries of Nature that we have been unable to explain simply trans­form into mere arti­facts of an out­dated, incom­plete map. With a dimen­sion­ally richer map we might just find the uni­fying res­o­lu­tion we’ve been searching for.

The Polish math­e­mati­cian Theodor Kaluza was one of the first to dis­cover the uni­fying poten­tial that comes with addi­tional dimen­sions. In 1919, while working at the University of Konigsberg in Germany, Kaluza took it upon him­self to rework Einstein’s for­mula for gen­eral rel­a­tivity after adding an extra dimen­sion to its struc­ture. When he did this he came up with extra equa­tions, which turned out to be the equa­tions Maxwell had written down for describing light. [2] Simply by assuming that the uni­verse con­tained an addi­tional space dimen­sion, Kaluza dis­cov­ered a math­e­mat­ical frame­work that com­bined Einstein’s equa­tions of gen­eral rel­a­tivity with those of Maxwell’s equa­tions of electro­magnetism (Figure 3-4). Of course Kaluza did not just pull this idea from thin air. He rec­og­nized that Einstein’s theory of rel­a­tivity had opened a door to the pos­si­bility of extra dimen­sions and he became curious about them.

[FIGURE PLACE HOLDER]

Figure 3-4 Maxwell’s and Kaluza’s equations.

Since Kaluza, sci­en­tists have become more and more con­vinced that addi­tional dimen­sions have the ability to sim­plify and unify the laws of Nature math­e­mat­i­cally. But a com­plete higher dimen­sional frame­work has not yet been con­structed. The reason behind this is largely related to the long held belief that humans are men­tally inca­pable of com­pre­hending a higher dimen­sional frame­work — that extra dimen­sions are impos­sible to visu­alize. [3] After all, our entire array of expe­ri­ences, from our very first moments in life to our most recent expe­ri­ences, rein­force a con­cep­tual model of three spa­tial dimen­sions. So if space pos­sesses more than three dimen­sions, how can we be expected to com­pre­hend the realms that are entirely beyond our experiences?

This ques­tion reminds me of the inqui­si­tions Michio Kaku made when he was a young boy. Sitting next to a pond in the Japanese Tea Garden in San Francisco mes­mer­ized by the bril­liantly col­ored carp swim­ming slowly beneath the water lilies he said:

In these quiet moments, I felt free to let my imag­i­na­tion wander; I would ask myself silly ques­tions that only a child might ask, such as how the carp in that pond would view the world around them. I thought, what a strange world theirs must be!

Living their entire lives in the shallow pond, the carp would believe that their ‘uni­verse’ con­sisted of the murky water and the lilies. Spending most of their time for­aging on the bottom of the pond, they would be only dimly aware that an alien world could exist above the sur­face. The nature of my world was beyond their com­pre­hen­sion. I was intrigued that I could sit only a few inches from the carp yet be sep­a­rated from them by an immense chasm. The carp and I spent our lives in two dis­tinct uni­verses, never entering each other’s world, yet were sep­a­rated by only the thinnest bar­rier, the water’s surface.

I once imag­ined that there may be carp ‘sci­en­tists’ living among the fish. They would, I thought, scoff at any fish who pro­posed that a par­allel world could exist just above the lilies. To a carp ‘sci­en­tist’, the only things that were real were what the fish could see or touch. The pond was every­thing. An unseen world beyond the pond made no sci­en­tific sense.

Once I was caught in a rain­storm. I noticed that the pond’s sur­face was bom­barded by thou­sands of tiny rain­drops. The pond’s sur­face became tur­bu­lent, and the water lilies were being pushed in all direc­tions by water waves. Taking shelter from the wind and the rain, I won­dered how all this appeared to the carp. To them, the water lilies would appear to be moving around by them­selves, without any­thing pushing them. Since the water they lived in would appear invis­ible, much like the air and space around us, they would be baf­fled that the water lilies could move around by themselves.

Their ‘sci­en­tists,’ I imag­ined, would con­coct a clever inven­tion called a ‘force’ in order to hide their igno­rance. Unable to com­pre­hend that there could be waves on the unseen sur­face, they would con­clude that lilies could move without being touched because a mys­te­rious, invis­ible entity called a force acted between them. They might give this illu­sion impres­sive, lofty names (such as action-at-a-distance, or the ability of the lilies to move without touching them).

Once I imag­ined what would happen if I reached down and lifted one of the carp ‘sci­en­tists’ out of the pond. Before I threw him back into the water, he might wiggle furi­ously as I exam­ined him. I won­dered how this would appear to the rest of the carp. To them, it would be a truly unset­tling event. They would first notice that one of their ‘sci­en­tists’ had dis­ap­peared from their uni­verse. Simply van­ished, without leaving a trace. Wherever they would look, there would be no evi­dence of the missing carp in their uni­verse. Then, sec­onds later, when I threw him back into the pond, the ‘sci­en­tist’ would abruptly reap­pear out of nowhere. To the other carp, it would appear that a mir­acle had hap­pened. After col­lecting his wits, the ‘sci­en­tist’ would tell a truly amazing story. ‘Without warning,’ he would say, ‘I was somehow lifted out of the uni­verse (the pond) and hurled into a mys­te­rious nether world, with blinding lights and strangely shaped objects that I had never seen before. The strangest of all was the crea­ture who held me pris­oner, who did not resemble a fish in the slightest. I was shocked to see that it had no fins what­so­ever, but nev­er­the­less could move without them. It struck me that the familiar laws of nature no longer applied in this nether world. Then just as sud­denly, I found myself thrown back into our uni­verse.’ (This story, of course, of a journey beyond the uni­verse would be so fan­tastic that most of the carp would dis­miss it as utter poppycock.)

I often think that we are like the carp swim­ming con­tent­edly in that pond. We live out our lives in our own ‘pond,’ con­fi­dent that our uni­verse con­sists only of those things we can see or touch. Like the carp, our uni­verse con­sists of only the familiar and the vis­ible. We smugly refuse to admit that the par­allel uni­verses or dimen­sions can exist next to ours, just beyond our grasp. If our sci­en­tists invent con­cepts like forces, it is only because they cannot visu­alize the invis­ible vibra­tions that fill the empty space around us. Some sci­en­tists sneer at the men­tion of higher dimen­sions because they cannot be con­ve­niently mea­sured in the lab­o­ra­tory.” (Kaku 1995, 3-5)

Like the carp of Kaku’s pond, we struggle to under­stand the causes of the ‘forces’ of our expe­ri­ence. It is our failure to imagine the dimen­sions beyond our imme­diate grasp that keeps us from seeing out­side of our own ‘pond’. If we wish to truly under­stand the nature of the uni­verse we inhabit, we must over­come our con­cep­tual blind­ness and learn to dis­tin­guish between what is inside and what is out­side of our space­time pond. In this way we will find that the mys­te­rious move­ments of our ‘lilies’ have simple explanations.

Although recently the leading drive behind the real­iza­tion of more dimen­sions has come from our search for an expla­na­tion for quantum mechan­ical effects, it is Einstein’s dis­cov­eries that actu­ally pro­vide us with the strongest con­cep­tual key that we can use to open our eyes to this higher dimen­sional realm. To wield this power we must follow the clues we pos­sess about space­time; we must con­sider the prop­erty of cur­va­ture that space­time man­i­fests and follow its rev­e­la­tions to our solu­tion. Let’s begin by asking some impor­tant questions.

If there are extra dimen­sions, then where are they? What direc­tions are orthog­onal to the direc­tions we have already described? How could there be spa­tial infor­ma­tion that is com­pletely inde­pen­dent of, or orthog­onal to, the familiar x, y, and z dimen­sions? How can it be pos­sible to move in a spa­tial direc­tion without moving in x, y, or z? What if one of these addi­tional dimen­sions is another time dimen­sion? When would it be? How could we visu­alize or com­pre­hend more dimen­sions in addi­tion to the ones we are familiar with?

As we con­tem­plate these ques­tions we must keep in mind the fun­da­mental def­i­n­i­tion of a dimen­sion. A dimen­sion pro­vides inde­pen­dent, orthog­onal spa­tial or tem­poral infor­ma­tion about phys­ical reality. Each dimen­sion maps the nat­ural realm in a com­pletely inde­pen­dent fashion. The infor­ma­tion that a fourth, fifth, sixth, and so on spa­tial dimen­sion would pro­vide must be entirely sep­a­rate from length, width, and height. Therefore, addi­tional spa­tial dimen­sions must express entirely new direc­tions. They must map aspects of posi­tion entirely sep­a­rate from x, y, or z. In short, in order for a para­meter to be a new space dimen­sion, it must be pos­sible to move about in that dimen­sion without moving in x, y, or z. Ultimately, this require­ment will be what allows us to defin­i­tively claim whether or not we have dis­cov­ered a new spa­tial dimen­sion. If we end up with a map that allows us to geo­met­ri­cally move without moving through the dimen­sions of x, y, or z, then we can con­fi­dently say that this motion takes place within an inde­pen­dent spa­tial dimension.

Many of us, including myself, were actu­ally taught that attempts to visu­alize more than three dimen­sions are futile because our brains are “inca­pable of com­pre­hending them.” This is absolutely not true! As we will soon dis­cover, (See part II) the pow­erful sym­metry of dimen­sional hier­archy allows us to simul­ta­ne­ously visu­alize more than three dimen­sions. Once we gain the ability to see eleven-dimensionally we gain intu­itive access to the secret work­ings of Nature. Since the sim­plicity of space­time itself exists in a realm of eleven dimen­sions we must tap into that full dimen­sional geom­etry to grasp its secrets. To help expose the missing parts of the map let’s begin a process of log­ical deduc­tion – a process that will non-arbitrarily intro­duce these extra dimen­sions and begin to reveal their form.

Curvature and Hidden Dimensions

The sim­plest obser­va­tional clues we have of higher dimen­sions come from our obser­va­tions of curved space­time. In order to account for the cur­va­ture of space­time while map­ping the uni­verse, we dis­cover that we must use at least seven inde­pen­dent vari­ables. For example, x, y, z, s, m, d, and t, where x, y, and z rep­re­sent orthog­onal spa­tial dis­tances from an origin, the Greek let­ters s (sigma), m (mu), and d (delta) rep­re­sent dimen­sions that enable depic­tion of the cur­va­ture pos­sessed by those three direc­tions, and t rep­re­sents time. [4]

Einstein attempted to depict the exis­tence of addi­tional dimen­sions by graph­i­cally sup­pressing a familiar spa­tial dimen­sion and drawing a dimen­sion that allowed him to rep­re­sent cur­va­ture in its stead. He used a visual rep­re­sen­ta­tion of a rubber sheet being stretched by a bowling ball. (Figure 3-5) The bowling ball rep­re­sents a mas­sive object, like a black hole or the sun, and the stretched mem­brane of the rubber sheet rep­re­sents a slice of spacetime’s reac­tion to the bowling ball’s presence.

The assump­tion that you cannot visu­alize more than three dimen­sions at once makes the use of this two dimen­sional rubber sheet nec­es­sary in order to graph cur­va­ture. For each familiar plane, one extra dimen­sion is nec­es­sary to describe its cur­va­ture. Therefore, for three planes (xy, yz, zx which can be thought of as two per­pen­dic­ular walls and the floor), three extra dimen­sions are required to explain the com­plete cur­va­ture of space. In order to account for the com­plete cur­va­ture of the (x, y, z) metric, three addi­tional dimen­sions are nec­es­sary. (For Figure 3-5, only one dimen­sion is nec­es­sary to rep­re­sent cur­va­ture because the spa­tial dis­tor­tion it is rep­re­senting is that of only one plane.) It should be clear that this model is not equipped to help us visu­alize the cur­va­ture of three space dimen­sions at once. (Not to men­tion the fourth dimen­sion of space­time, which is time.)

This model pos­sesses other short­com­ings which only con­fuse our ability to explain the nature of space­time. You might look at this dia­gram and ask: is the weight of the bowling ball causing the rubber sheet to stretch? If this cur­va­ture is used to explain gravity, then is it not a cir­cular expla­na­tion to visu­ally use the weight of the bowling ball, which is a func­tion of gravity, to describe the cause of curvature?

Figure 3-5 A slice of space­time warping into another dimension.

Is gravity the cause of gravity? This rep­re­sen­ta­tion is unsat­is­fac­tory because it leaves us with no sense of what actu­ally causes this warping of space­time. Furthermore, if this dia­gram is sup­posed to help us under­stand the warping of space­time, how does it give us any rep­re­sen­ta­tion of warped time? Not only does the rubber mem­brane offer an analogy that allows us to visu­alize just a thin slice of space, it also offers no expla­na­tion for the warping of time.

If you are familiar with these kinds of rep­re­sen­ta­tions of curved space­time then you might have noticed that this figure con­tains some­thing dif­ferent in com­par­ison to the stan­dard fig­ures. What’s dif­ferent is that the dimen­sion that space­time is warping into is actu­ally labeled. Traditional rep­re­sen­ta­tions like this, for some reason, fail to label this other dimen­sion — leaving it unmen­tioned com­pletely. But it is absolutely vital to remember that this often-unlabelled dimen­sion is the very dimen­sion that enables depic­tion of the cur­va­ture of our space­time plane in the first place. Its pres­ence in our pic­to­rial expla­na­tion should not be ignored or over­looked — espe­cially when our goal is to under­stand the com­plete picture.

In order to develop a model that is capable of graph­i­cally demon­strating the com­plete cur­va­ture of space­time, without sup­pressing any of the familiar space dimen­sions or ignoring time, let’s examine what is meant by curvature.

Imagine that we have an obser­va­tion sta­tion on Earth and that we have placed three obser­va­tion sta­tions in space (in the con­fig­u­ra­tion shown in Figure 3-6a). This gives us four unique observers. If we task all four observers with the job of con­tin­u­ously mea­suring the loca­tion of the new­born star Dilabee while also mon­i­toring the posi­tions of the other three observers their jobs will be quite boring. As they watch, day after day, they see no changes. All four sta­tions agree that there is no mea­sur­able velocity between any of the observers or Dilabee. Therefore, the rel­a­tive posi­tions between these five objects all remain con­stant and the geo­metric con­fig­u­ra­tion of the group is static.

One year, how­ever, some­thing con­founds this whole set up. Someone on Earth notices a black hole with a com­panion star (which makes the posi­tion of the black hole easy to mea­sure) trav­eling on a path that will bring it between Earth and Dilabee. Strangely, as the black hole moves closer and closer toward a posi­tion that will place it between Earth and Dilabee, Earth’s observers see Dilabee’s posi­tion changing —ending up with an increased dis­tance and a new angle with respect to the angle it was pre­vi­ously observed. (Figure 3-6b)

Figure 3-6 (a, b) Effects of a black hole.

The star’s posi­tion appears to change from Earth’s point of view when the black hole comes into the pic­ture, but the three obser­va­tion sta­tions do not detect any change.

When the observers on Earth examine the three sta­tions in space they detect no change in their posi­tions, so they ask the three space sta­tion observers to verify that Dilabee has changed posi­tion. The three space sta­tions all agree that The Earth-based observers are wrong. They see no change in the star’s angle or dis­tance. From their per­spec­tives Dilabee has not moved at all.

This effect is real. It is some­thing that has been detected and mea­sured many times over by sci­en­tists around the world. Einstein came up with a geo­metric way to describe this effect. (Note that the example I have used thus far has been two-dimensional. That is, the four observers and the star all lay in the plane of the paper. In Nature this effect is not lim­ited to two dimen­sions.) If we rotate the plane of the dia­gram, sup­pressing the third familiar spa­tial dimen­sion, and replacing it with a dimen­sion that is out­side of familiar space, then we can ‘see’ Einstein’s graph­ical descrip­tion of this effect by allowing cur­va­ture to be drawn in, and there­fore pic­to­ri­ally expressed by, this unfa­miliar third dimen­sion. (Figure 3-7) From this we can see that the cur­va­ture caused by the black hole accounts for the per­ceived changes in dis­tance and direc­tion. The three space sta­tions do not yet detect any change because they are still in rel­a­tively ‘flat’ space. They are not observing the star through curved or warped spacetime.

Figure 3-7 Curvature explains black hole effects. The way the black hole curves or warps space­time into another dimen­sion explains why the observers from Earth see the star change posi­tion while the obser­va­tion sta­tions do not.

This brings us to at least a par­tial expla­na­tion (visual descrip­tion) of curved space­time. A par­tial expla­na­tion is better than no expla­na­tion, but wouldn’t it be nice to acquire a com­plete depic­tion of curved space­time. Wouldn’t it be won­derful to be able to visu­alize this cur­va­ture without having to sup­press one of the familiar dimen­sions? If only we could figure out how to do this. If only we could figure out how to extend Einstein’s expres­sion of curved space to include the missing spa­tial dimen­sion of our expe­ri­ence? While we are at it, we might also want to shoot for a depic­tion that is also capable of revealing curved time. To do this we need to know exactly what this cur­va­ture is.

It is impor­tant to remember that Einstein’s dia­grams of warped space­time are expres­sions of a varying char­ac­ter­istic of space­time from one region to another. In our example with Dilabee the depic­tion is meant to convey that there is actu­ally more space between Earth and the charted star when a mas­sive object lies between them. Depiction of cur­va­ture is a schematic rep­re­sen­ta­tion of that increase in amount of space. The slope of this cur­va­ture por­trays how steeply the changing spa­tial mea­sures depend upon prox­imity to a mas­sive object. With this under­standing let’s con­sider a volume of space and inquire about this effect we call cur­va­ture. We will draw what I believe are impor­tant con­clu­sions from this exercise.

Imagine that our volume of space is defined as a cube. (Figure 3-8a) Let’s say that at each of the eight cor­ners of the cube there exists an observer. These observers com­mu­ni­cate their dis­tances to each other con­tin­u­ously. Every mea­sure­ment made between the observers finds that they are in com­plete agree­ment about their fixed posi­tions, and that they have zero rel­a­tive veloc­i­ties. Every observer mea­sures the dis­tances to each of the other cor­ners. For example, A mea­sures the dis­tance to B and C and finds that they are 90° from each other and of equal dis­tance. Through simple geom­etry observer A can deter­mine the dis­tance that B and C will mea­sure between each other:  times the dis­tance between A and B. Each of the eight observers can mea­sure the dis­tances to the other seven observers and can then cal­cu­late the dis­tances that each of the other obser­va­tion sta­tions will record for their mea­sure­ments. All of these cal­cu­la­tions and mea­sure­ments exactly agree.

Now, if we place a black hole near the center of this cube, (Figure 3-8b) what will we find? We will dis­cover that when the observers mea­sure dis­tances along the edges or the faces of this cube they still mea­sure the same dis­tances and posi­tions that they mea­sured before. Because of this, those observers will expect that the dis­tances con­necting the fur­thest cor­ners of the cube will be iden­tical to what they mea­sured before. But, when observers actu­ally mea­sure the dis­tances between cor­ners whose line passes through the center of the cube, as from C to D, they find that there is more space between them. C sees D as spa­tially more dis­tant and dis­placed com­pared to how it appeared before the black hole was intro­duced. D also sees C as being fur­ther away and dis­placed. In fact, all dis­tances have increased for mea­sure­ments that survey paths that come within prox­imity of the black hole. [5] Strangely, how­ever, the volume of space, defined by the posi­tions of the eight cor­ners, has remained the same.

Figure 3-8 (a, b) Surveying volume.

(a) All cal­cu­la­tions and mea­sure­ments of the dis­tance between the eight cor­ners agree.(b) Measurements that uti­lize paths close to the black hole no longer agree with cal­cu­la­tions. They always increase.

What does this mean? How can you have ‘more space’ in the same volume? The answer turns out to be quite pro­found and yet sur­pris­ingly simple. Just as tri­an­gles are no longer defined by a total of 180° in curved space, the vol­umes of cubes can vary with cur­va­ture. To visu­alize the curved tri­an­gles (two dimen­sional objects), we simply place them on a curved sur­face. (Figure 3-9) But how can we visu­alize three dimen­sional curved objects or regions? The answer makes the pos­tu­la­tion of extra hidden spa­tial dimen­sions no longer arbi­trary or extrav­a­gant. It is this ques­tion that ini­ti­ates a chain of deduc­tive rea­soning that leads to the inescapable phys­ical exis­tence of extra dimensions.

[FIGURE PLACEHOLDER]

Figure 3-9 Triangles on a sphere and a saddle.

To answer this ques­tion let’s turn to a more familiar example. Let’s take two cubes of equal size. One is made of dia­mond and the other is made of graphite. Both, there­fore, con­tain only carbon. If these cubes are painted black we may guess that they are in every way iden­tical since we are told that they are com­posed of the same mate­rial. But upon picking both cubes up we will quickly sur­mise that one is heavier than the other. How can one explain this? They are of equal volume and are both made of only carbon, so how could their weights vary?

Naturally we turn to a descrip­tion of den­si­ties. We explain that the cubes are made of ‘atoms’ or small mass par­ti­cles, in this case carbon atoms. Inside the dia­mond cube these par­ti­cles are packed more closely together than they are in the graphite cube. (Figure 3-10) In other words, the dia­mond cube is denser than the graphite cube. This is why they can be com­posed of the same mate­rial, have the same vol­umes, but pos­sess dif­ferent masses or weights.

Diamond

Graphite

Figure 3-10 Lattice struc­tures of dia­mond and graphite.

When carbon is sub­jected to a pres­sure under 20,000 atmos­pheres it takes on the crys­talline lat­tice struc­ture of graphite. Over 20,000 atmos­pheres it takes on the crys­talline lat­tice struc­ture of diamond.

It is impor­tant to rec­og­nize that in order to explain the con­cept of den­sity we must describe, or com­pre­hend, two things: the par­ti­cles (atoms) and the medium (space) in which the par­ti­cles are dis­trib­uted. If we did not assume or visu­alize the medium (space) in which the atoms reside, then we could not explain vari­able densities.

In our example of the spa­tial cubes, we find the same sit­u­a­tion. Two cubes of equal size that con­tain a dif­ferent amount of ‘stuff.’ Only this time the ‘stuff ’ we are refer­ring to is space itself. Therefore, when we think through the impli­ca­tions of curved space, we find that cur­va­ture is a descrip­tion of how regions of space are not uni­form. It expresses that equal vol­umes can con­tain dif­ferent amounts of space. In par­tic­ular, regions near mass con­tain more space than regions far from mass. This real­iza­tion urges men­tion of vari­able den­si­ties for space, which in turn strongly sug­gests that space itself is par­tic­u­late and that its pieces are dis­trib­uted within a medium that serves as even a back­ground to space. Since this medium enables the dis­per­sion of the pieces of space it must pos­sess spa­tial dimen­sions that are entirely sep­a­rate from the dimen­sions held by those pieces. This is where we will find the addi­tional dimensions.

Where does this leave us? Well, so far we have found that a uni­verse made of space­time that has the prop­erty of cur­va­ture lends itself to a line of deduc­tive rea­soning that guides us to the con­clu­sion that the fabric of space­time is com­posed of dis­crete quantum packets. These quantum units of space can be arranged with vari­able den­si­ties. The punch line is that this con­di­tion requires the lit­eral exis­tence of extra dimen­sions. Therefore, through a quantum model of space­time, extra dimen­sions are no longer arbi­trary assump­tions, extrav­a­gant pos­tu­la­tions, or inspired guesses — they are nec­es­sary con­clu­sions. The quan­tized nature of space­time requires them. Because of this, the next step in our quest to attain a com­plete map of phys­ical reality shall be to explore this quan­tized struc­ture of space­time. We shall begin by exam­ining other clues from the micro­scopic realm that lead to this finding, and then we will con­sider the prop­er­ties of these indi­vidual pieces of space.

[Continue to Chapter Four]

From the forth­coming book:

Einstein’s Intuition
by Thad Roberts

Represented by
Sam Fleishman
Literary Artists Representatives
New York, New York

NOTES:

[1] Remember that inde­pen­dent para­me­ters, or bits of infor­ma­tion, must be telling us some­thing about the metric of phys­ical reality. In other words, they must relate to ‘where’ or ‘when’ an event occurs. Color, as it turns out, is some­thing that is already encoded in the metric when we include all the dimen­sions. At any rate color doesn’t tell us any­thing about where or when some­thing is.

[2] The added spa­tial dimen­sion was posited as cir­cular. This is an impor­tant point that will come into play later. Kaluza had pro­duced five extra quan­ti­ties. Four of these could be used to pro­duce Maxwell’s elec­tro­mag­netic equa­tions. Walter Isaacson, Einstein.

[3] Even today’s promi­nent physi­cists have a dif­fi­cult time wrap­ping their minds around higher dimen­sions. They seem to have pre­ma­turely jumped onto the ‘impos­sible’ band­wagon — claiming that because they haven’t visu­al­ized higher-dimensional realms, it must be impos­sible. For example, in his recent bookHyperspace, Michio Kaku reflected the cur­rent ten­dency to accept this impos­si­bility when he said: “How do we see the fourth spa­tial dimen­sion? The Problem is, we can’t. Higher dimen­sional spaces are impos­sible to visu­alize — so it is futile even to try.” Michio Kaku, A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (New York: Anchor Books, 1995).

Stephen Hawking con­curs with the words, “It is impos­sible to imagine a four-dimensional space. I per­son­ally find it hard enough to visu­alize three-dimensional space!” (Hawking, ‘A Brief History Of Time,’ p. 24.) In Lisa Randall’s opinion: “It’s not thinking about extra dimen­sions but trying to pic­ture them that threatens to be unset­tling. Trying to draw a higher-dimensional world inevitably leads to com­pli­ca­tions.” (Lisa Randall,Warped Passages)

Dr. Randall believes in the phys­ical exis­tence of addi­tional dimen­sions; she just doesn’t think it is pos­sible to visu­alize them along side the familiar dimen­sions. This atti­tude has very strong roots in his­tor­ical phi­los­ophy. The modern meta­phys­ical tragedy that clings to this almost unan­i­mously accepted claim can be summed up in Immanuel Kant’s (1724-1824) con­clu­sion that “since we lack direct access to ‘reality in itself,’ we are lim­ited to what we per­ceive.” (Diane Barsoum Raymond, Existentialism and the Philosophical Tradition.) The pre­sup­posed inability to visu­alize or con­cep­tu­alize higher-dimensional maps com­pletely forces this conclusion.

All of this echoes the sen­ti­ments of Werner Heisenberg who set the tone for modern physics with the claim that we should “abandon all attempts to con­struct per­cep­tual models of atomic processes.” (Werner Heisenberg, ‘Physics and Beyond,’ New York, Harper & Row, 1971, p. 76.)

Despite the his­tor­ical failure to do so, visu­al­izing higher dimen­sional realms is not impos­sible. With the right insight it is actu­ally quite simple. It is accom­plished through real­izing levels of dimen­sional hier­archy by allowing the fabric of space­time to be com­posed of stip­pled con­stituents residing within a volume of super­spa­tial dimen­sions. By the end of this book we will all be able to visu­alize more than three spa­tial dimen­sions simultaneously.

[4] Notice that I did not choose (x, y, z, i, j, k, t) despite quantum mechan­ical laws demanding the exis­tence of ‘imag­i­nary’ dimen­sions for each spa­tial dimen­sion. This is because the dimen­sions that allow depic­tion of cur­va­ture are in fact sep­a­rate from the dimen­sions tied to ‘imag­i­nary’ char­ac­ter­is­tics, which describe quantum mechan­ical sys­tems. Compactified ver­sions of these ‘imag­i­nary’ dimen­sions will come into play later.

The three Greek let­ters s, m, d (low­er­case sigma, mu and delta,) were taken as pho­netic com­po­nents of the Sanskrit word Samadhi.

[5] Light sig­nals also take longer to travel through regions with high cur­va­ture because they have to tra­verse more space. This is known as the Shapiro effect.