Keeping the Fourth

Now and then, and for some reason especially of late, I find myself stuck on (not necessarily in) the fourth dimension. (By “4D” I mean Euclidean 4-space, not spacetime which is also four-dimensional but very different). Thinking about it is a fun time-killer actually, if you’re a bit visuospatially inclined—even in the inevitable failure to really picture the tesseract or the Clifford torus, say, there is a fascination in the notion that what we see in front of us may not be all there is even to its appearance. When an added invisible dimension comes into play, even such simple familiar things as rotation seem deceptive: they may, in fact, produce changes that violate everything we intuitively know about how “real” shapes are supposed to act.

There is a kind of lush, baroque appeal to the thought of an unseen continuation of the space that we think we know so well, a continuation which may contain who knows what wonders or terrors. The idea of additional dimensions almost perfectly captures the sense of mysterious “beyondness”, the urge to expand or to reach for new depth and ultimately for infinity, that particularly entrances the imagination of the West, as evident in so much of its religion, art, science and technology. Therefore, it is little surprising that the concept of the fourth dimension has inspired reams of literature, notably in the 19th and early 20th century, and even has become something of a trope, including such works as Abbott’s 1884 “Flatland”, H.G. Wells’ “The Time Machine”, Madeline L’Engle’s “A Wrinkle In Time”, H.G. Lovecraft’s “The Color Out of Space”, and Nelson Bond’s “The Monster from Nowhere”. In the visual arts, the most famous beneficiary of four-dimensionality might be Cubism, which aimed to represent all sides of a 3D object simultaneously, much as a 4D being might.

It is interesting that most such works have undertones, not only of mathematics or science-fiction or pure entertainment, but also often the occult—of secret or privileged knowledge. Indeed an additional dimension would be “occult” in the most literal sense, as the word is originally from the Latin occultus, for “secret”, or “unseen”. Hidden yet inescapable, the fourth dimension, even if it does not exist, offers a powerful modern metaphor for the occult, both in its possibility of shocking new truths and powers and in its implicit suggestion that all that one needs to gain such powers is to change one’s sight, to turn one’s gaze in a direction that had been there all along but was simply neglected by the ignorant masses. Higher hidden dimensions can also be a metaphor for other phenomena that nevertheless have the flavor of divination-by-other-means, for instance in Schopenhauer’s dimensionally-suggestive remark about the nature of genius (“talent hits a target no one else can hit; genius hits a target no one else can see“), or the very common descriptions of artistic inspiration as “otherworldly” in origin or “from another plane”.

Furthermore, it is commonly argued or implied in literary instances of the fourth dimension that four-dimensional beings would possess powers and understanding of our world that would appear to us as nearly indistinguishable from the godlike. In our three-dimensional world, they could appear and disappear at will out of apparent nothingness; they could read the insides of sealed objects as we read the pages of a book; they could achieve seeming-impossible transformations of matter and structure. This only adds to the curiously mystical aura which naturally accretes around thoughts of higher dimensions. In short, it is as though four-dimensional space affords the inmates of our scientific, rationalism-bound culture an intellectually respectable venue for belief in the supernatural (or supranatural)—after all, the fourth and higher dimensions are, quite by definition, out of this world.

And yet, to a mathematician, the concept of four-dimensional or much higher or even infinite dimensional spaces is a completely hard-headed and even prosaic one, and the mystic accretions are of little interest, mere “woo”. The problem with this matter-of-fact attitude seems to be that the now-prosaic step of combining a fourth spatial dimension with the three we “know” did not become a widely used or understood procedure in mathematics until sometime in the mid-nineteenth century. Was this perfectly prosaic device of the fourth dimension and higher spaces then “discovered”, or “invented”—or something else?


No less fascinating than the creative or pseudo-mystical possibilities opened up by thinking about the fourth dimension is the way it exercises our philosophical attitudes about the dimensions we know—or think we know. For if we seriously consider the matter, and make an honest attempt to understand how we would convert our intuitive conception of space to include this extra dimension, the extreme difficulty of doing so starts to erode even the mathematical confidence of what “space” is—that it is really equivalent to the Cartesian product of straight lines at right angles to each other, and so on. Instead the mood turns increasingly Kantian, as it dawns that space as we know it may not be the mathematician’s space at all, but an irreducible, inaccessible category of our understanding, for which certain mathematical formalisms only happen to give a very good model. Those formalisms also happen to allow many fruitful conceptual extensions, as mathematical concepts so frequently do—but nothing more; in Kant’s words, space “…is nothing but the form of all appearances of outer sense. It is the subjective condition of sensibility, under which alone outer intuition is possible for us” (COPR, 71). Space “…is not a discursive or, as we say, general concept of relations of things in general, but a pure intuition” (COPR, 69).

To the extent that I can tell from my own experience and what I glean of others’, the interesting truth is that we do not ever really think in terms of “three” dimensions as we are going about our business in the world—this is an abstraction, a model, after the fact. The process of identifying space with a Cartesian grid made of orthogonal left-right, up-down, and in-out directions (or spherical, or cylindrical, or any other triplet of coordinates) requires considerable abstraction, if not training in geometry. And so when we face the fourth dimension, and try to picture 3D space “as we know it”, except now with a specifically added direction, the notion of appears intuitively bizarre and arbitrary, despite that the concept seems coherent and the math works.

This is not to say that the arising of the number “three” when we apply said abstraction to our experience is not fascinatingly mysterious—only that direct intuition refuses to underwrite the extensions and variations that the mathematics, charging ahead, insists are completely “natural” and even “real”. Reams of bewildered questions spill forth. What is it, this extra dimension, taken outside of mathematics? Why is it, why should it “be” there, as part of actual space, even in our imagination? What does it even mean for a dimension to “be” or “not be”? What is this arbitrary “adding” process supposed to be, if the Cartesian product of a line with our familiar 3-space fails to jibe with anything our intuition can call space? And even if I do successfully “add” another dimension, what is to tell me that I added it in the right way—is there some way that a completely new direction would not be at a right angle to the others? Should it be a small looped dimension or an infinite straight one, and how do I tell other than just assigning it to be one or the other? Where, after all, is this other dimension? How do we make it “attach” to the other three—and if it fails to attach, “where” does it go?

Telling oneself instead something like “and now in addition to this 3D, let there be another thing that I can vary continuously” allows a certain amount of intuition to seep in, but it is emphatically not a spatial understanding in the sense of space as we daily and constantly perceive it, since this again makes one axis “odd man out” as an “another thing”, when the lived experience of “space” implies complete integrality, free rotation and movement, again, with no choice of or even attention usually given to axes, coordinates, etc.

Alternatively, one can try to “paint” a kind of intuitive spatial quale, a raw feeling of “closeness” or “farness”, onto a 3D region to indicate different hyper-depths, but this still has the strange effect of singling out the fourth dimension, making it an add-on instead of fully equivalent to the other three or, what is really desired, intuitively not a specific “dimension” at all, but fully natural percept of “four-space”. Particularly, this approach fails on the condition of rotation—if we think of our added fourth dimension as something like “temperature”, say, we realize as soon as we try to turn our heads to face “temperature-wise” that there is a problem, as trying to rotate a real object in real space into the temperature axis is completely nonsensical (even to the greatest synesthete). Yet mathematically this can be accomplished easily—many kinds of data analysis like PCA we may choose new axes without much concern for the original units.

What is worse, as soon as we begin to ask these questions about the fourth dimension using the mathematical imprimatur, the way is open to begin decomposing and doubting our own natural space, which becomes perforce “3D”. We are haunted now not only by the aforementioned mystery of “why three dimensions (and not some other)”, but also of how the x, y, and z axes “fit” together and whether there is some kind of spatial “glue” that keeps the parallel planes from sliding over each other and creating a mess, like graphite. Appeals to lower-dimensional analogy in general, as in the famous 2D “flatland” stories, are typically presented as the road to proper understanding in the direction of increasing dimension, but in the direction of decreasing dimension we find they do not really get us where they are intended to at all. For “Flatland” aside, in truth to us “2D life” is as inconceivable and artificial an idea as 4D (or, for that matter, 3D) life. To exist on a plane of zero thickness and perfect flatness is, as far as we can tell, almost as nonsensical a concept as “turning to face temperature”—and we cannot in fact “picture” anything of zero or infinitesimal thickness—so what are we supposed to gain by the analogy?

Maybe the most interesting thing about attempts to intuitively visualize four-space is the way that the experience starts to resemble that of pondering many of the ancient conundrums of philosophy. One begins to feel one has made progress, only to realize that one has been assuming the answer all along, or else making blatant mistakes. Could the fourth dimension, like these intractable philosophical riddles, be nothing more than one of the linguistically-created “spirits”, born of “a tendency to sublime the logic of our language”, which Wittgenstein warned us about? Or is the difficulty we face in intuiting such higher spaces simply an example of how, to quote Fodor in The Modularity of Mind, “it is surely in the cards that there should be some problems whose structure the mind has no computational resources for coping with”?

Chomsky, often a fellow-traveler of Fodor’s in these kinds of arguments, often invokes a little tableau involving a Martian who, watching human philosophers fight endlessly about the nature of free will, is astonished that the humans simply never hit upon the way to answer the question—an answer which, given the Martian’s distinctive mental structure, is perfectly obvious. Visualization of the fourth dimension perhaps is analogous, in which case the fault is in our brains. But what is difficult about this response is that it implies a kind of Platonism—for it insinuates that the “problems”, with their solutions, must exist in some sense even when there is no one around with the right brain modules to conceive of them. The truth about free will, or 4D visualization, is out there, waiting, so that other beings than us, endowed with the right mental capacities, can catch hold of it. In that case, however, what distinguishes the 4D space that already “exists” out there, waiting to be experienced—say by some lucky Martian—from the 4D space that appears not to exist, based on the best evidence we have from the world around us?

Faced with this barrage of questions, the semi-mystic function of “higher spaces” or “the beyond” as a placeholder and metaphor for a missing understanding serves us all too well; for to the question of where these wondrous Platonic verities might be found, we may as well suggest, however saucily: perhaps they themselves are in another dimension!


For those interested in experiencing the unique feeling of 4-dimensional bewilderment for themselves (in case the above wasn’t enough), it’s hard to beat the iPhone app “Rotation4D”—which, for a puny 2.6 megabytes of memory space, lets you watch very nicely rendered color projections of the 4-dimensional regular polytopes (the four-dimensional equivalent of the Platonic solids) and set them rolling through various kinds of higher-dimensional rotations at the flick of a finger. There is a hypnotic beauty in seeing the different constituent shapes or “cells” of the polytope materialize out of the “hidden” fourth dimension, twirl into view, and then slowly flatten as they tilt out of our 3D “hyperplane” and fade back out of sight—and it’s probably as good a way as any to approach an intuitive understanding of 4D.

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