The Impossibility of the Fourth Dimension

Tesseract rotating in the fourth dimension (Hise, 2006).

The principle of 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences' would seem to point to the reality of the Fourth Dimension. So why don't we perceive this Fourth Dimension?

Let's talk for a moment about Newton's laws of motion. He managed to concoct a single mathematical description that explained two entirely different orders of phenomena: a planet orbiting its star and an apple falling from a tree.

In 1960, physicist Eugene Wigner wrote an article titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences in which he remarked on the almost magical manner in which mathematics serves as the ideal language for formulating the laws of physics. 

Beyond Newton's laws, another common example are Maxwell's equations, which provide the basis for classical electrodynamics, classical optics, and electric circuits. These serve as the foundation for all electrical, optical and radio technologies. 

"The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it," Wigner wrote.

It seems that it was this same sort of awareness of the mystical nature of mathematics that inspired Plato to create his World of Forms.

In his paper, Wigner noted that a physical theory's mathematical structure frequently provides clues to further advances and can even lead to empirical predictions. Plato would love to hear that. Plato was the master of making predictions from theory though he never cared much for empirical confirmation.

The magic of mathematics is that it can be used to create laws which seem to exist in a reality somehow higher than that of the physical objective world.

Platonist Henry More was the first to use the term "Fourth Dimension" though he used it to refer to Plato's World of Forms and not a fourth spatial dimension. Still as we have seen, Plato is responsible for both the idea of alternate dimensions and that of a spatial dimension beyond the usual three.

It was 1637 when Descartes published his work on what would come to be known as the Cartesian Coordinate System. This system merged geometry with algebra and led to the development of calculus.

On one hand, people were fascinated by the correspondence between the three dimensions of physical objective reality and the way they were so perfectly captured by the cubic cells of the Cartesian Coordinate System. On the other hand, mathematicians were making calculations in four dimensions. These calculations are conducted in four-dimensional Euclidean Space. These facts, taken together, seemed to indicate that there must be a spatial fourth dimension somewhere beyond our ability to perceive.

In 1827, as I noted in The Gnostic Notebook: On Plato, The Fourth Dimension and the Lost Philosophy, August Ferdinand Moebius wrote: 

"… two equal and similar solid figures, which are however mirror images of each other, could be made to coincide if one were 'able to let one system make a half revolution in a space of four dimensions. But since such a space cannot be conceived, this coincidence is impossible in this case."

Moebus had already taken the first few steps into the Fourth Dimension. In 1880, W.I. Stringham’s published an article titled, "Regular Figures in N-dimensional Space", in the American Journal of Mathematics. In this article he included sketches of various higher dimensional shapes, including the tesseract.

First published diagram of a hypercube (Stringham, 1880, Plate I).

Even though we cannot perceive the Fourth Dimension, we can formulate mathematical rules to determine what features these higher dimensional shapes possess. 

Orthographic projection of hypercube (Ruen, 2010).

For instance, as the next diagram makes clear, we begin with a single point at the zero-dimensional level. At the one-dimensional level we have a line, but more importantly we have doubled the number of points to two. At the two-dimensional stage we again have a doubling of points from two to four with the square. This doubling continues with the square's four points becoming the cube's eight points. Finally we arrive at the Fourth Dimension, the level of the hypercube or the tesseract, which no one has ever actually seen, yet we can determine from our mathematical progression that this figure will have double the number of points when compared to the cube.

Dimension levels (NerdBoy1392, 2008).

There is also the mathematical equation known as Euler's Formula. This formula, when applied to four-dimensional shapes, is V - E + F - C = 0 where V = number of vertices, E = number of edges, F = number of faces, and C = number of (3-dimensional) cells.

Let's look at the hypercube:

Vertices are points, so V = 16. Edges are lines. A cube has 12 lines, two cubes have 24 lines and then there are 8 additional lines connecting the two cubes, so E = 32. Faces are planes. These are easier to see if we render the hypercube in the following manner:

Hypercube (Mouagip, 2010).

The outer cube has 6 faces, the inner cube has 6 faces and connecting these two cubes are 12 additional faces. So F = 24. Finally we come to the value of C which is the number of three-dimensional cells. There is the cell at the center and the cell of the larger cube giving us 2 and then there are cells between each of the two cubes' faces, making 6 cells, so C=8.

V - E + F - C = 0 

or

16 - 32 + 24 - 8 = 0

The math works out. Does this mean that a fourth spatial dimension actually exists? 

Henri Poincaré wrote in his The Value of Science (1907) :

“… experience does not prove to us that space has three dimensions; it only proves to us that it is convenient to attribute three to it.”

And in his Science and Hypothesis (1905) he suggested: 

“A person who should devote his existence to it might perhaps attain to a realization of the fourth dimension.”

This opened the door to any and all who wished to theorize, promote, or attempt to reach the Fourth Dimension. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics as did charlatans, psychics, mystics, and spiritualists. 

Salvador Dali's 1954 Crucifixion (Corpus Hypercubus).

In 1875, astrophysicist Johann Karl Friedrich Zöllner became interested in Fourth Dimension after being exposed to Spiritualism. With the assistance of the medium Henry Slade, Zöllner set up a series of seance experiments which included the interlinking of two wooden rings, slate-writing, tying knots on string, and retrieving coins from sealed boxes. He eventually came to the conclusion that the spirits were able to accomplish these feats because they are four-dimensional entities. Later, it turned out that the medium Henry Slade was actually a talented fraudster who had deceived Zöllner. No matter, the idea that spirits existed within the Fourth Dimension was accepted as fact by a gullible public.

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" principle would seem to point to the reality of the Fourth Dimension. Note however that this principle doesn't say that all mathematical formulations mirror actual physical relationships. Rather it says that those mathematical formulations which do mirror actual physical relationships seem to do so too perfectly.

But, even as the hypercube and the Fourth Dimension were growing in popularity within the public imagination, another mathematical based theory of dimensions was rising in importance. In this theory, however, the fourth dimension was not a spatial dimension but was rather the dimension of time and this space-time was not found in four-dimensional Euclidean space but rather in non-Euclidean Minkowski space.

Deformation of spacetime caused by a planetary mass (Mysid, 2015).

It was this version of the Fourth Dimension with its dimpled rubber sheet of space-time that won out over the idea of a fourth spatial dimension. Let's not forget that the failed idea of a fourth physical dimension leads all the way back to Plato. It is a result of theorising from logic and theory without concern for empirical confirmation. Though the idea of a fourth spatial dimension was superseded by the reality of space-time, the ideas of parallel dimensions and higher dimensional realities continued to fester within the underground, only to have reemerged within the public's consciousness during the last few years.

In my next post, we will examine two strands of this underground belief system that have somehow grown together and become stronger during their years within the shadowy realms of the public's unconscious mind: the ceremonial magick system devised by Aleister Crowley and the Cthulhu Mythos of H. P. Lovecraft.

Aleister Crowley 1912

H. P. Lovecraft 1934

Plato, the Father of the Fourth Dimension

Plato. Luni marble, copy of the portrait made by Silanion ca. 370 BC for the Academia in Athens.

Plato was the father of both Alternate Dimensions and the Fourth Dimension. The otherworldly nature of these two binds them together like the single side of a Moebius Strip.

Readers of my book, The Gnostic Notebook: On Plato, the Fourth Dimension, and the Lost Philosophy, will recall the various connections between the works of Plato and the 'discovery' of the Fourth Dimension. These connections include the Analogy of the Cave with its depiction of prisoners entranced by the shadows cast on their cave wall. The prisoners mistake the shadows for the real three-dimensional world. In just this simple description we have two key techniques for the analyzing of higher dimensions: Projection, which is related to the casting of shadows, and Dimensional Analogy which is where we imagine how a two-dimensional being would experience a three-dimensional object as in Edwin Abbott Abbott's Flatland.

It was also Plato who, with the formation of his academy at Athens, set into motion the collection and development of mathematical and geometrical knowledge which resulted in Euclid's Elements. This led, after a couple of thousand years, to the creation of the Cartesian coordinate system. 

Plato was also the first to come up with the idea of an alternate dimension. He called his the World of Forms and identified it as a dimension filled with eternal and unchanging archetypes such as Truth, the perfect circle, the number 2, and the Pythagorean theorem. 

The point of having the World of Forms is to be able to able to have a realm of absolutes outside of the material realm. That way one can recognize what is good because one has knowledge of the Good from the World of Forms and then one can find that which resembles the Good in the material realm.

This was all in an attempt to get to the essence of a thing. When we speak about something, we are using words to describe the thing. These words are themselves descriptions of aspects of the things, but how do we know that these other words correctly capture the essence of the aspects they describe? The deeper we go into examining the terms, the more obvious it becomes that these terms are just words used by those who came before us, none of which can bring us any closer to the thing's essence. Hence the need to jump out of the Universe to the realm of absolutes where the unchanging truth dwells. 

Before Plato, the entire concept of a separate alternate dimension didn't exist. God was on top of a mountain, Hades was beneath the earth. 

After Plato came the Christian heaven and hell, realms of eternal bliss or eternal suffering. Eternal because they exist outside of the material realm and therefore outside of time and space.

It seems as though these three aspects of Plato's philosophy: the dimensional aspects of the Analogy of the Cave, the alternate dimension of the World of Forms, and his support of research in mathematics and geometry which culminated in Euclid's Elements, apparently led to both the 'discovery' of the Fourth Dimension and the proliferation of alternate dimensions. 

Moreover, the concepts of the Fourth Dimension and Parallel Dimensions are closely intertwined. We live in a space of three dimensions. Up - Down, Left - Right, and Forward - Back. Just like the citizen of Flatland who cannot perceive the Up - Down dimension, it may be that there is a hidden fourth perpendicular along which we can move. If so, that may mean that these parallel dimensions are quite close to our own when we move along this fourth dimension, though they remain unreachable from the usual three.

Using this sort of reasoning, it soon becomes obvious that Heaven and Hell must exist within the Fourth Dimension. As to the existence of the Fourth Dimension itself, we will consider that impossibility in the next post.

Dimensions Beyond the Usual Three

The idea of parallel Universes and additional hidden spatial dimensions is a common feature in pop culture as well as in areas of scientific research such as quantum mechanics and Superstring Theory, but is there any actual empirical data supporting their existence?

The concept of dimensions beyond the usual three pervades our culture. In a recent episode of the Netflix series, Stranger Things, the following exchange takes place at a funeral home after one of the main characters has apparently been put to rest:

Mike, Lucas, and Justin question Mr. Clarke at Will's wake.

Mike: So, you know how in Cosmos, Carl Sagan talks about other dimensions? Like, beyond our world?

Mr. Clarke: Yeah, sure. Theoretically.

Mike: Right, theoretically.  

Lucas: So, theoretically, how do we travel there? 

Mr. Clarke: You guys have been thinking about Hugh Everett's Many-Worlds Interpretation, haven' you? Well, basically, there are parallel universes. Just like our world, but just infinite variations of it. Which means there's a world out there where none of this tragic stuff ever happened. 

Lucas: Yeah, that's not what we're talking about. 

Mr. Clarke: Oh.

Dustin: We were thinking of more of an evil dimension, like the Vale of Shadows. You know the Vale of Shadows? 

Mr. Clarke: An echo of the Material Plane, where necrotic and shadow magic...

Mike: Yeah, exactly. If that did exist, a place like the Vale of Shadows, how would we travel there?

Lucas: Theoretically.

Mr. Clarke explaining the Acrobat and the Flea.

Mr. Clarke: Well... Picture an acrobat standing on a tightrope. Now, the tightrope is our dimension. And our dimension has rules. You can move forwards, or backwards. But, what if right next to our acrobat, there is a flea? Now, the flea can also travel back and forth, just like the acrobat. Right? 

Mike: Right.

Mr. Clarke: Here's where things get really interesting. The flea can also travel this way along the side of the rope. He can even go underneath the rope.

Boys: Upside down.

Mr. Clarke: Exactly.

Mike: But we're not the flea, we're the acrobat. 

Mr. Clarke: In this metaphor, yes, we're the acrobat. 

Lucas: So we can't go upside down?

Mr. Clarke: No.

Dustin: Well, is there any way for the acrobat to get to the Upside Down?

Mr. Clarke: Well you'd have to create a massive amount of energy. More than humans are currently capable of creating, mind you, to open up some kind of tear in time and space, and then you create a doorway. 

Dustin: Like a gate?

Mr. Clarke: Sure. Like a gate. But again, this is all...

Lucas: Theoretical.

Mike: But, but what if this gate already existed? 

Mr. Clarke: Well, if it did, I, I think we'd know. It would disrupt gravity, the magnetic field, our environment. Heck, it might even swallow us up whole. 

In the scene above Mr. Clarke touches on two types of alternate dimensions. The first is Hugh Everett's Many-Worlds Interpretation, which, according to Wikipedia, states that "there is a very large—perhaps infinite—number of universes, and everything that could possibly have happened in our past, but did not, has occurred in the past of some other universe or universes."

The second type discussed by the boys and Mr. Clarke are parallel planes of existence, and one in particular: "An evil dimension, like the Vale of Shadows," according to Dustin. 

So we have two types of alternate dimensions. The first is a scientific theory, the second a commercialized mockery of an esoteric tradition, while the means to bind the two together within the narrative, the acrobat and the flea analogy, is a simplification of a single aspect of Superstring Theory. So there are actually three varieties of alternate dimensions being referenced in this single scene: Everett's Many-Worlds Interpretation, Esoteric Planes of Existence, and finally Superstring Theory, which itself requires a total of ten dimensions of space-time.

Let's see what the skeptics say concerning just the scientific extra-dimensional theories:

"For a start, how is the existence of the other universes to be tested? To be sure, all cosmologists accept that there are some regions of the universe that lie beyond the reach of our telescopes, but somewhere on the slippery slope between that and the idea that there are an infinite number of universes, credibility reaches a limit. As one slips down that slope, more and more must be accepted on faith, and less and less is open to scientific verification. Extreme multiverse explanations are therefore reminiscent of theological discussions. Indeed, invoking an infinity of unseen universes to explain the unusual features of the one we do see is just as ad hoc as invoking an unseen Creator. The multiverse theory may be dressed up in scientific language, but in essence it requires the same leap of faith."  Paul Davies, A Brief History of the Multiverse

Concerning string theory, according to Wikipedia, theoretical physicist Lee Smolin, in his 2005 book The Trouble with Physics,

"claims that string theory makes no new testable predictions; that it has no coherent mathematical formulation; and that it has not been mathematically proved finite." Also "string theory has yet to come up with a single prediction that can be verified using any technology that is likely to be feasible within our lifetimes."

If we write off Multi-World Theory as fantasy and Superstring research as being pure theory devoid of any empirical verification, that leaves us with just three dimensions of space and one of time, along with perhaps another four dimensions of momentum space, but most importantly with only the three observable dimensions of space. 

Finally we come to the other sort of alternate dimension which was discussed within the scene, that most persistent of delusions: the Parallel Plane of Existence, which in this instance is a realm labeled the Vale of Shadows or The Upside Down.

Concept art from Stranger Things.

Readers of my book, The Gnostic Notebook: On Plato, The Fourth Dimension, and the Lost Philosophy will recognize Plato's World of Forms as both the archetypal prototype of and the dualistic opposite to "the Vale of Shadows, an echo of the Material Plane, where necrotic and shadow magic" stem.

In my next post, we will delve further into the nature of these esoteric parallel planes of existence.