The Fourth Dimension
If a fourth dimensional creature existed it could, in our three dimensional universe, appear and dematerialize at will, change shape remarkably, pluck us out of locked rooms and make us appear from nowhere. It could also turn us inside out. (Carl Sagan)
...if there are three dimensions, a real body of two dimensions cannot exist... In the same way...if there are four dimensions, a real body of three dimensions cannot exist... If the fourth dimension exists while we possess only three, it means that we have no real existence, that we exist only in somebody's imagination... If we do not want to agree with this we must recognize our-selves as beings of four dimensions. (P.D. Ouspensky)
During the transition from the so-called "classical" era of physics to the quantum weirdness of today, a visionary briefly appeared dazzling everyone with his "free inventions of the mind". His work was treated with respect, admiration and awe and was given volumes of generous lip service. For instance, "As the creation of a single mind, it is undoubtedly the highest intellectual achievement of humanity."1 Yet, after extracting what was useful, science has refused to acknowledge or understand his revelations and what they may imply. Max Born said in 1955 why this may be so:
(Notice the carefully chosen phrase "physical intuition". To Einstein, as with many of us, intuition was no more "physical" than were statistical probabilities. Born, as we have already seen, is very generous, if not enlightening, with his use of the term "physical".)
Relativity is the theory of relationships. And its connections with experience are certainly not "slender", as we shall see. It is by far the most profound, far reaching, irrefutable--and thus--stubborn, deeply entrenched thorn in the development of the new physics. From top to bottom, it is almost completely opposite from the quantum theory. For instance:
Relativity is geometric, and thus graphic. Quantum theory is statistical and allows few graphic pictures.
Relativity is deterministic: "an effect is a measure of its cause and consistency is a measure of truth" (Poe). Quantum theory is based on uncertainty (indeterminacy), allowing random effects.
Relativity is based on a continuum, where the interval between any two points can be divided infinitely. Quantum theory has a built in limitation: the discontinuous, indivisible quanta.
Relativity is deductive, with necessary conclusions. But quantum theory is inductive, requiring observation/measurement and some very creative interpretations.
Relativity covers the world at large – the entire spectrum – while quantum theory covers only the very small.
Relativity is logically flawless, where quantum theory is logically absurd and so denies logic!
Relativity has one author: Einstein. Quantum theory has many authors, including, Einstein.
Relativity has no need for the quantum. Quantum theory must use relativity. And so on...
Obviously, this comparison shows as drastic a duality conflict as that of the wave and the particle. There is, however, one major point in common between this strained marriage. And that is that both must acknowledge the existence of the fourth dimension in order to work successfully. Relativity demands it. Quantum theory reluctantly defers to it. But what does this say about the actual existence of the fourth dimension to us mortals?
It appears therefore more natural to think of physical reality as a four-dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.
...there is no more commonplace statement than that the world in which we live is a four-dimensional space-time continuum. (Albert Einstein)
The scene of material events, the world, is therefore a four-dimensional metrical continuum. (Herman Weyl)
...the real three dimensional world is obsolete and must be replaced by the four dimensional space-time with non-Euclidean properties... The four dimensional world is no mere illustration; it is the real world of physics, arrived at in the recognized way by which physics has always (rightly or wrongly) sought for reality. (Sir Arthur Eddington)
What this means is that not just some distant arena of abstract space is four dimensional, but physical objects as well.
There seems to be a physical law that objects capable of physical existence must have as many dimensions as the surrounding space. (Hans Reichenbach)
These particles can be conceived only in relativistic terms, that is, a framework where space and time are fused into a four-dimensional continuum... Subatomic particles must be conceived as four-dimensional entities in space-time. (Fritjof Capra)
To accurately describe electrical and mechanical phenomena in a manner valid for all observers, physicists have to visualize the world in space-time, not in space plus time. (Nick Herbert)
I belabor the point because most people are not aware as to what extent, if any, the fourth dimension really exists. And this is because science itself is confused about this very issue.
Talking about the fourth dimension is like talking about God. We must assume its existence in one form or another, but we don't know much about it. No two descriptions are quite the same. We create models, but they always come up short. We form images but eventually they become incompatible with expanding knowledge. We project all kinds of idealistic and unrealistic attributes on these images only to find that they just reflect our combined hopes, fears, prejudices and, ultimately, just plain ignorance.
There is one major difference however. Science must accept the fourth dimension but emphatically rejects God: "God is a forbidden word in science."3 And while much of its efforts have been directed towards the de-mystification of God (and other uncomfortable ideas including mysticism itself), it shows no misgivings about mystifying the fourth dimension:
It is not possible for us actually to see the geometry of the space-time continuum because it is four-dimensional and our sensory experience is limited to three dimensions. For that reason, it is not even possible to picture it.4
This aptly describes the general consensus on the matter. Others, however, go even further, adding insult to injury. In a statement concerning science fiction writers, some of our most creative and learned citizens (many are even scientists), the authors say:
...the imaginations of science fiction writers are simply too limited to grasp the true features of higher dimensional universes given to us by rigorous mathematics.5
(The audacity of this statement reflects the audacity of the title of the book it was taken from.)
The point is that the mysterious and unfathomable nature of the fourth dimension is an unwarranted assumption and an unnecessary self-delusion. It comes from the refusal to accept what reasoning dictates and the logical consequences derived from it. But mostly...the implications.
Many writers on the subject equate us with two dimensional "Flatlanders"; shadow beings who have discovered that a higher dimension exists but lack the imagination and ability to look up into the third dimension. By this allusion, we lack the physical and mental capabilities to perceive, or even conceive of, the fourth dimension. And so we are told repeatedly, and at times with excessive hubris, that "it can only be understood mathematically."
But this is non-sense. First of all, "rigorous mathematics" has nothing whatsoever to do with governing the limits of the imagination, nor intelligence, nor the fourth dimension. It is an intellectual detour, a disguise and a ruse. We're talking about physical reality here and we're certainly not flatlanders; nor do we live in "higher dimensional universes" – which is science-fiction.
Secondly, a first year geometry student could easily construct a four dimensional figure such as the 4-d hypercube, or tesseract shown in figure 1. This figure is derived logically and geometrically by extending a ninety degree perpendicular (right angle) from the sum of all prior dimensions. This cube within a cube is the only model of a four dimensional cube that can be produced using "normal" geometric consistency and the defining characteristic noted above. For this is the same criteria used to graphically produce the other dimensions. And with one exception, it is the model used most often in popular expositions on the subject.
Carl Sagan says about this model:
But for a real tesseract, in four dimensions, all the lines would be of equal length and all the angles would be right angles.6
But this is faulty logic. The lines on both cubes are all equal in length. For the exception noted above is that we have placed a measuring rod on the bottom front line of each cube. Each line, in fact, could be considered a measuring rod. If our world is four dimensional then everything in it must be four dimensional including measuring rods and clocks. And the two cubes are a representation of one cube that has evolved in space and time. Therefore the measuring rod must evolve proportionally. Why Sagan ignored this has always been a mystery to me.
On the second objection above, all angles are right angles. For right angles are defined as ninety degree perpendiculars. And in the evolution of dimensions, these perpendiculars always extend from the sum of all preceding dimensions. To get the sum, simply place the evolving figure on a plane surface so that all preceding dimensions (lines) correspond to the plane equally (fig. 5-6). The plane surface then becomes the sum. And the line extending ninety degrees from the sum travels through the opposite diagonal corner (fig. 6).
So then, what exactly is the fourth dimension? Sir Arthur Eddington, the first to verify the general theory of relativity tells us:
Density multiplied by volume in space gives us mass, or what appears to be the same thing, energy. But from our space-time point of view, a far more important thing is density multiplied by a four-dimensional volume of space and time; this is action... Action is the curvature of the world.7
The term "action", in this sense, comes from electrodynamics and thermodynamics and represents motion. More explicitly, it means least motion in both space and in time. Thus action is motion; is the fourth dimension; is the relationship between both cubes; is the four-dimensional, space-time "interval", and thus is the four dimensional space-time continuum.
Now the lines between the two cubes represents this action, the fourth degree of freedom of motion; an infinite number of transmuting volumes in space and time. The theory of relativity allows us to measure these lines by using both rods and clocks. But it can only be a virtual measurement. For at this stage, an actual measurement has meaning only from an imaginary and relative ("privileged") point of reference; that is, from the G.O.D. But, and this is the whole point, it can be imagined.
The idea of the fourth dimension ought to have arisen from observation of a series of progressively growing or diminishing spheres or cubes. One of the clearest and most comprehensive forms of motion in the fourth dimension in this sense is growth, the principle of which lies in expansion.8 (Ouspensky)
In other words, the entire concept of the fourth dimension–the volumes of complex rhetoric and "rigorous mathematics" in "higher dimensional universes"–can be reduced to three simple words: growth by expansion.
The fourth dimension is simply growth (or decrease) in size, a most natural process, or long awaited "mechanism", that can only be seen by the imagination. It's the movement in or out from either cube to the other. And since the four dimensional action involved is in space and over time, the fourth dimension is not just time, creating some non-visual, mysterious, space-time continuum, but it is the space-time continuum itself. The confusion lies in the semantics which reflect explanations of what nobody has heretofore wanted to accept. Because just like the spreading wave function, the implications of this conclusion are astounding. And just like the spreading wave function, its unanimous denial is based, not on a logically sound argument, not by empirical verification, and certainly not by any experimental results, but entirely on its profound implications.