GATEWAY TO THE
PART I: THE HYPERCUBE AS THEATER
In the first entry on this Fourth Dimension web page, we met the basic figure of four-dimensional geometry, the hypercube (or tesseract, as it’s often known). We watched as a familiar, quite ordinary three-dimensional cube began extending itself into a strange new dimension, to emerge before our eyes as a fully four-dimensional entity. Now we turn to an elegant representation of this same figure in the form of a model, and ask ourselves an obvious question: how can such a four-dimensional entity be replicated in a model of a mere three dimensions?
We begin with the model in question (for larger versions of the model just ckick on the image):
Its resemblance to the hypercube, albeit with its roof removed and a few important inner parts omitted, seems apparent. It reveals the same fourfold symmetry, inherited as we saw from an origin in the cube. Yet I think we must be careful about the term model. If “model” means a small-scale version of the real thing, this can’t be a model of the hypercube! This is evidently a photograph of some built thing and no four-dimensional entity can be squeezed into a three-dimensional space.
Counting plane faces and allowing for the missing roof, this is a regular solid of twelve square faces and Euclid, whose Elements culminate in a conclusive inventory of all regular solids, does not allow for this one! (His own regular solid of twelve faces, the dodecahedron, has pentagons for faces.) A more incisive characterization of the hypercube, based on a more complete image, is that of a figure with six faces, all of which are cubes. But the verdict is the same: there is no room for this figure in a space of a mere three dimensions. There simply isn’t elbowroom in our entire universe for the small object in the photograph!
What, then, can it be? It must be a “model” in some other sense.
I propose that our figure is indeed a model, only not of the hypercube per se. Suppose we were to read this as a model of the 3D visual image formed as we look directly at a 4D hypercube?
Two objections arise immediately:
(1) We can’t see entities in the fourth dimension, so how could there be a “visual image” of such a thing? and (2), even if we could see a four-dimensional figure, our human vision is itself always two-dimensional, not three. Our retinas are only two-dimensional, are they not?
Let’s consider the second objection first, since it appears decisive. Indeed, it is unfortunately quite true that our eyes are incapable of forming an optical image of even the 3D entities which surround us in the world we inhabit. In that sense, the objection is valid: we cannot see even the common cube! Yet we resolutely affirm that we certainly do see it. Our concept of seeing evidently goes far beyond the literal forming of an optical image. We clearly work around this problem by using our intelligence, learning from infancy how to triangulate on the images from our pair of eyes, and incorporating the evidence from multitudes of informal visual experiments. Like the denizens of Abbott’s Flatland, confined to life in a table top for whom a straight line is a wall blocking the view of anything beyond it we too are stuck in an inadequate space, contriving to think our way around this problem as best we can. What we call seeing is a construct which goes far beyond the optical evidence with which it begins.
With this response to the second objection, we answer the first as well. Just as we say with conviction that we see a cube though we know that optically speaking, we do not so in the same sense we may combine intelligence with limited optics, and learn as well to see things in the fourth dimension. Our visual image of the cube is already a construction; why might we not extend these interpretive strategies to see the hypercube, as well?
Our model, then, is not of the hypercube, but of the visual image of the hypercube, viewed along a specific line of sight perpendicular to the page on which the photograph is reproduced. As in the case of the cube, we would of course see something quite different along a different line of sight, and this would require the construction of a different model. Our figure has emerged as indeed the buildable thing it appears to be, the 3D model of a particular 4D thing, the hypercube. We may thus claim full visual access to the hypercube and our model reveals what it looks like!
Beyond this, however, we can now draw a much broader conclusion. This specific image can become an instrument, or as we might say, a stage, for viewing all other four-dimensional figures. We haven’t called attention to the two crossed sticks, prominent in our original image. These are diagonals of the hypercube, intersecting at the center of the hypercube’s space. If the hypercube is to be a stage on which other entities may appear, we need to host them with some common coordinate frame. Let Ω serve as center of that frame, origin of a four-dimensional coordinate system. All four axes of the four-dimensional system will pass through it, and our hypercube will provide a complete coordinate frame with respect to within which any figure in the 4D world can be placed and oriented. With this accomplished, our hypercube has indeed become a theater, upon whose stage all the figures and events of this new world can make their appearances.
The analogy to theater is indeed striking. We have emphasized that we see our hypercube from only one, specific point of view; if we wanted to change viewpoint, we would need to make another model. But this is precisely the situation we occupy in any theater: we take our seats, and the world appears before us battlefields, shipwrecks or soliloquies. In fact, our theater bears a striking likeness to Shakespeare’s Globe Theater, which he at one point refers to as his “wooden O”. Our theater, too, is globular. If sticks were run from Ω to its corners, they would all prove equal projections of a hypersphere! Indeed, we are in a theater, occupying balcony seats of the perfect line of sight, front-row-center!
Everything capable of existing in 4D is now open to our view. But we must not, of course, fall into Don Quixote’s disastrous mistake, confusing one domain with another, going to battle with the puppets! Our set is three-dimensional, and gives us only what that space is able to offer: bold perspective, for example, light and color, the attributes of theater -- but never a look behind the scenes. If we want to see one of our actors from the side, for example, we’ll have to build another set to house him in. As we recognized from the outset, three dimensions cannot house even the tiniest grain of four-dimensional, quadric sand! Keep this single rule in mind, and our hypercube theater offers unbounded visual access to everything four-dimensional, waiting only to be conceived and displayed!
Let us reap the reward of this reasoning by inviting one representative, altogether four-dimensional entity to make a cameo appearance on our stage.
This bird-like figure happens to be the graph, in four coordinates, of the complex parabola an entity which many solemn mathematicians to this day have asserted could not be drawn! Since graphing of several variables is one of the life-lines of our modern, deeply interconnected world, perhaps this graph of four interrelated quantities, exotic as it appears, may suggest that our theater project may have practical, as well as esthetic value.
With this birth of our theater, we have accomplished only half of our overall project, which is to establish the hypercube as “Gateway” to the four-dimensional world. We have seen entities appear, but we need also to be able to create them. Our model-maker has been busy depicting the joining of two hypercubes and once he has shown us that, all other constructions of 4D design will become open to us. Part II, on the design and construction of 4D entities, is expected to appear in the near future. Stay tuned!
This study is conducted in preparation for a book,
Models, photos and art work in this article by Eric Simpson.