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