Chapter 3

Section 3: Extra Dimensions

If there really are other dimensions in the fabric of physical reality, then we ought to attempt to gain ontological access to those dimensions, to learn how to see them. That task has the poential to unmask the outstanding mysteries of Nature, to reveal those mysteries as mere artifacts of an outdated, incomplete map.

The Polish mathematician Theodor Kaluza affirmed 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 3-3).



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 medium of our universe, inventing ‘forces’ to make up for the shortcomings of our picture. But with the right picture we might find that the mysterious movements of our ‘lilies’ have simple explanations.

If extra spatial dimensions exist, 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 x, y, and z? How can it be possible to move in a spatial direction without moving in x, y, or z? What if one of the 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 let’s keep in mind that a dimension provides independent, orthogonal information about the spatial or temporal structure of physical reality. Each dimension maps the natural realm in a completely independent fashion. Additional spatial dimensions must provide information that is entirely separate from length, width, and height (x, y, z). They express entirely new directions.

If we propose a new spatial dimension, it must be possible to move about in that dimension without moving in x, y, or z. Recall that z is an independent dimension because it is possible to move in z without moving in x or y. We must meet this requirement if we are to definitively claim that we have unveiled a new spatial dimension. If we end up with a map that allows geometric movement that doesn’t involve movement through 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 taught that attempts to visualize more than three dimensions are futile because our brains are “incapable of comprehending them.” In this book I will outright defy this claim. We will discover that we are capable of simultaneously visualizing more than three spatial dimensions (see part II). And when we do we will gain greater intuitive access to the secret workings of Nature (see part III). To reach that end goal, let’s begin a process of logical investigation – a process that will non-arbitrarily introduce these extra dimensions and begin to reveal their form.


Curvature and Hidden Dimensions


Clues of higher dimensions come from our observations of curved spacetime. In order to account for the curvature of spacetime while mapping the universe, it might be necessary to 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, t represents time, and the Greek letters s (sigma), m (mu), and d (delta) represent dimensions that enable depiction of the curvature possessed by those three directions. [4]

To elaborate on this point, note that Einstein pictorially made use 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 3-4) 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.


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


The assumption that you cannot visualize more than three dimensions at once, combined with the assumption that space is a continuum, 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 3-4, 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 well equipped to help us simultaneously visualize the curvature of three space dimensions (not to mention the fourth dimension of spacetime – time.)

This model possesses other shortcomings that 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 and therefore gravity?

Is gravity the cause of gravity? This representation is unsatisfactory because it leaves us with no sense of what is actually responsible for 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 fails to offer us an intuitive connection to why time warps in regions of curvature.

If you are familiar with these kinds of representations for curved spacetime, then you might have already noticed that Figure 3-4 slightly differs from the standard figure. What’s different is that the dimension that spacetime is warping into is actually labeled (with a question mark). Traditional representations of spacetime curvature usually fail to label this other dimension — leaving it unmentioned completely. But this often-unlabeled dimension is the very dimension that enables depiction of the curvature of our spacetime plane. 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 3-5a). The four unique observers continuously measure the location of the newborn star Dilabee while also monitoring the positions of the other three observers. 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.




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.


One day, however, something confounds this whole set up. A black hole with a companion star (which makes the position of the black hole easy to measure) comes into the picture, traveling on a path that will bring it between Earth and Dilabee. Strangely, as the black hole moves closer and closer toward a position between Earth and Dilabee, Earth’s observers see Dilabee’s position changing — its distance increases and its angle changes. (Figure 3-5b)

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.

Although this example has been fabricated 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 3-6) 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. Our goal is the complete that picture. To do that we need to examine what curvature is.

Spacetime curvature represents a physically real characteristic of the vacuum whose value varies smoothly from one region to another. Depiction of curvature is a schematic representation of a change in the ‘amount of space’ or a change in metric density from one region to the next. The slope of this curvature portrays the magnitude of change at each point, which depends upon the proximity of that point to massive objects and the magnitude of their masses. 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 with a side two astronomical units long (an AU is the distance between the Sun and Earth). (Figure 3-7a) At each of the eight corners of the cube there exists an observer. These observers keep an open line of communication about their distances. They are all in complete agreement about their fixed positions and zero relative velocities. 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 measure the distances to the other seven observers and 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 a black hole drifts into the center of this cube, (Figure 3-7b) what changes? Well with the black hole at the center the observers measure the same distances along the edges or the faces of the cube. Because of this, those observers may expect that the distances connecting the furthest corners of the cube will also be identical to what they measured before. But, when they 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 a significant increase. There is far more space between opposing corners. All distances have increased for measurements that survey paths that come within proximity of the black hole. [5] There is ‘more space’ inside this cube. Strangely, however, the volume of space, defined by the boundary of the cube, 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 when curvature is introduced. To visualize the curved triangles (two dimensional objects), we simply place them on a curved surface. (Figure 3-8) 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 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 kind of atoms. 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 3-9) In other words, the diamond cube is denser than the graphite cube. This is why they can be composed of the same kind of atoms, have the same volumes, but possess different masses or weights.





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.

If this line of reasoning holds up then spacetime curvature might be more richly explained if we assume that the fabric of spacetime is composed of discrete quantum packets. This is the seminal foundation upon which I propose to build my theory of quantized space, which I call quantum space theory (qst). Spacetime quantization requires the literal existence of extra dimensions. If spacetime is quantized, then extra dimensions are logically necessary instead of just being arbitrary assumptions, extravagant postulations, or inspired guesses. In Part II of this book, we will explore the full dimensional structure for spacetime that is suggested by quantization. To prepare us for that, we will now examine some of the clues that have popped up from the microscopic realm – clues that support the hypothesis that spacetime truly is quantized.



[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



[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.

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