Joyous Catastrophe: Flatland, Spaceland, and the Quest for Higher Dimensions

(Written 2007)

If it is possible that there are extensions with other dimensions, it is also possible that God has somewhere brought them into being; for His works have all the magnitude and manifoldness of which they are capable.
--Immanuel Kant, "Thoughts on the True Estimation of Living Forces"

In 1884 the English scholar and clergyman Edwin Abbott Abbott (1838-1926) published Flatland, a slim tome of mathematical science fiction. It tells the story of A. Square, a two-dimensional being (from Flatland, a vast Euclidean plane) who is introduced to the third dimension. Besides being an apt caricature of Victorian social mores, Abbott's book is a useful analogy for us modern readers as scientists posit higher spatial dimensions and we try to understand them. By approaching the third dimension from the perspective of the second, Flatland anticipates the problems of envisioning what mathematicians call "Hyperspace"--the realm of higher dimensions. [1] On yet another level, the novel can be read as spiritual quest: in his book The Fourth Dimension, Rudy Rucker writes, "A. Square's trip into higher dimensions is a perfect metaphor for the mystic's experience of higher reality." With Flatland's example in mind, this essay will attempt to explain for the layperson some mathematical, scientific, and spiritual implications of the fourth dimension, as well as address the underlying problem of the concept: how can we know anything about a dimension whose existence we cannot empirically prove?

It is probably best to begin, as Abbott does, with mathematics. We can easily enough understand the relationship between algebraic exponentiation and geometry: three to the second power (3²) equals nine, just as a square with a length and width of three inches has an area of nine square inches. Three cubed is twenty-seven, just as a cube with a length, width and height of three inches has a volume of twenty-seven cubic inches. Carrying this one step further, we know that three to the fourth power equals eighty-one; ergo, the four-dimensional analogue of a cube--what mathematicians call a tesseract--would have a length, width, height and "extra-height" of three inches and a hypervolume of eighty-one hypercubic inches. [2] Unfortunately, our brains are unable to understand such a hyperspacial object except by analogy: we know that a tesseract would be made up of cubes in the same way that a cube is made up of squares, but we find it extremely difficult to properly visualize it. [3] Furthermore, we cannot assume that, since a tesseract is algebraically possible, it is geometrically possible; the usefulness of four-dimensional equations is such that if the fourth dimension did not exist, it would be necessary to invent it. Though hyperspacial forms exist as a Platonic ideal, mathematics alone cannot tell us whether they are indeed real in our physical world. We can only, as A. Square does, "cast [ourselves] in faith upon conjecture, not knowing the facts."

Having failed to gain a concrete understanding through mathematics, we can attempt to comprehend the fourth dimension via the scientific hypothesis that necessitates its existence, String Theory. In The Elegant Universe Brian Greene likens our understanding of a multidimensional universe to than of a person viewing a suspended garden hose from far away. From a distance, the hose appears to be a one-dimensional line, though in fact its surface is two-dimensional: it has length and a circular dimension, which is "curled up", making it difficult to detect without a closer look. The fourth dimension Greene hypothesizes is both compactified to an infinitesimally small degree (far beyond anything we can currently detect), and located at every point in the three spatial dimensions we know of. For string theory to achieve its harmonious melding of general relativity and quantum mechanics, we must assume that the "strings" (one-dimensional structures considered by string theorists to be the fundamental components of reality) vibrate in at least nine spatial dimensions. But what are these dimensions? The theory, formulated by physicists Eugenio Calabi and Shing-Tung Yau, is that there is a shape with six dimensions compactified into it (a "Calabi-Yau space") at every point in the universe. When we move through space, we are actually moving through these higher dimensions; however, they're so minutely small that we don't notice them. Abbott, remarkably prescient as he was, noted something like this. In his preface to the second edition of Flatland he addresses the objection that Flatlanders must actually have an infinitesimally small third dimension of height, or they would be unable to see each other. A. Square responds,

"It is true that we have really in Flatland a Third unrecognized Dimension called 'height', just as it is also true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call 'extra-height'. But we can no more take cognizance of our 'height' than you can of your 'extra-height'. ... I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith."

String theory appears to be a fine resolution of some physics problems raised in the last century, but it is not verifiable; everything new it theorizes happens at such a small scale that we can't empirically prove it. We are left, like the Flatlanders, to make a leap of faith: we are no closer to a tangible confirmation of hyperspace than we were with mathematics.

Rational pursuits having failed us, we may as well examine some supernatural explanations concerning the fourth dimension. In Flatland, the Sphere, as a three-dimensional being, can reach through the fabric of Flatland to move tablets from one room to another, and can even touch A. Square's innards (much to the latter's distress). For the spiritually inclined it is tempting to postulate that what we call supernatural phenomena could somehow be the work of four-dimensional beings operating on our plane of existence. At first glace this appears to be a convenient solution for many of our deepest problems: the beings and concepts we once labeled as supernatural are simply a part of another dimension, with its own empirically deducible laws. The notion that our souls are hyperspacial forms dates at least as far back as 1659, when the Cambridge Platonist Henry More made the claim in his book The Immortality of the Soul. Unfortunately, More's assertion is little more than idle speculation, and it was completely forgotten for several centuries. In the second half of the nineteenth century there was rampant conjecture about the fourth dimension, encouraged both by mathematical speculation and by a growing fascination with the occult. Charlatans exploited this popular interest to explain paranormal phenomena performed at séances, calling on four-dimensional beings to do parlor tricks. As respected a scientist as Johann Carl Friedrich Zöllner, an early pioneer in the field of photometry, fell for such claims, zealously defending them. Con artists notwithstanding, the idea that the fourth dimension is a metaphysical realm was indeed popular among spiritualists; in 1888, A.T. Schofield wrote,

"We conclude, therefore, that a higher world than ours is not only conceivably possible, but probable; secondly, that such a world may be considered as a world of four dimensions... Though the glorious material universe extends beyond the utmost limits of our vision, even artificially aided by the most powerful telescopes, that does not prevent the spiritual world and its beings, and heaven and hell being by our very side."

However, if we accept the assumption that spiritual concepts are merely four-dimensional concepts, what about higher dimensions? If we accept the faith claim that one more spatial dimension exists, we are in no position to deny the possibility of additional ones. A truly omniscient, omnipotent, omnipresent God (or, at least, the notion of such a being) cannot be limited to a finite dimension, and thus we still have the problem of infinite regress; hyperspace cannot provide us with a neat and tidy mathematical solution to metaphysical questions.

Even were there a set number of dimensions, there are other problems with labeling God as a higher-dimensional being. A. Square, upon learning of the third dimension, goes further, envisioning a "Thoughtland"--we may read it as the dimension of infinity--where a being could look down on all dimensions and comprehend their forms. We may assume that a being in Thoughtland is God, but Abbott argues otherwise: "This omnividence [4]... does it make [one] more just, more merciful, less selfish, more loving? Not in the least. Then how does it make [one] more divine?" Divinity, for Flatland's author, is not a function of power and knowledge, but of righteousness; since there is nothing intrinsically virtuous about hyperspace, God cannot be God simply because he is a denizen of some higher dimension.

Ultimately, it seems that there is no union of theoretical mathematics with theology that can satisfy both mathematicians and theologians. Abbott, a clergyman himself, dismissed the effort to link mathematical understanding with spiritual improvement; in The Kernel and the Husk he wrote, "Even if we could conceive of Space of Four Dimensions... we should not be one whit better morally or spiritually. ... [N]o knowledge of Quadridimensional space can guide us [towards spirituality]". If we cannot use the sciences to further ourselves spiritually, then, conversely, spirituality will not help us advance our understanding of science; to conflate the two is to devalue both. In our condition, hyperspace can exist as a mathematical abstraction and object of speculation, but nothing more.

Our search for a fourth spatial dimension, then, must necessarily be a failure. Though we have the tools of mathematical, scientific, and spiritual inquiry, they cannot reveal hyperspace to us while we remain in our three-dimensional universe. We, the residents of Spaceland, have as much trouble envisioning Hyperspace as the Flatlanders have conceptualizing our world; while mathematical analogy is useful, it cannot truly show us because our brains are hard-wired, as it were, to grasp only the three spatial dimensions to which we are accustomed. Our scientific theories, though they are a fine basis for predicting physical phenomena, are not laws, and can only hypothesize hyperspace's existence. Likewise, spirituality can serve a valuable purpose, but it is not useful in the pursuit of the fourth dimension. Unlike A. Square, we have no otherworldly guide who has deigned to enlighten us, and thus must remain in our ignorance as long as we remain in this universe of ours.

If you liked this essay, you just might enjoy:

Abbott, Edwin Abbott. The Annotated Flatland. Ed. Ian Stewart. Cambridge, Massachusetts: Perseus Publishing, 2002.
Hinton, Charles. Fourth Dimension Writings.
Krauss, Lawrence M. Hiding in the Mirror. New York: Viking, 2005.
Rucker, Rudy. The Fourth Dimension. Boston: Houghton Mifflin, 1984.

Footnotes

2. There is a problem of language here. There is no accepted name for the direction of the fourth dimension, so I use "extra-height", a term used by Abbott. (An alternate word, devised by Henry More in the seventeenth century, is "spissitude".) "Hypervolume" and "hypercubic" are simply the four-dimensional equivalents of our words volume and cubic. [back]

3. It is worth noting that many have tried. A fine example is Robert A. Heinlein's short story "--And He Built a Crooked House--", in which an architect constructs a house in the shape of an unfolded tesseract; after an earthquake, the house somehow falls into the fourth dimension. [back]

4. "Omnividence" may be Abbott's own word, as a quick search reveals few sources besides him that use it. It simply means "the capacity of seeing all things", from the Latin OMNI "all" + VIDERE "to see". [back]