![]() ![]() Let Ω serve as center of that frame, origin of a four-dimensional coordinate system. If the hypercube is to be a stage on which other entities may appear, we need to host them with some common coordinate frame. These are diagonals of the hypercube, intersecting at the center of the hypercube’s space. We haven’t called attention to the two crossed sticks, prominent in our original image. This specific image can become an instrument, or as we might say, a stage, for viewing all other four-dimensional figures. We may thus claim full visual access to the hypercube – and our model reveals what it looks like!īeyond this, however, we can now draw a much broader conclusion. Our figure has emerged as indeed the buildable thing it appears to be, the 3D model of a particular 4D thing, the hypercube. 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 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. Our visual image of the cube is already a construction why might we not extend these interpretive strategies to see the hypercube, 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. With this response to the second objection, we answer the first as well. What we call seeing is a construct which goes far beyond the optical evidence with which it begins. 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. 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. Our concept of seeing evidently goes far beyond the literal forming of an optical image. In that sense, the objection is valid: we cannot see even the common cube! Yet we resolutely affirm that we certainly do see it. 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. Let’s consider the second objection first, since it appears decisive. Our retinas are only two-dimensional, are they not? (2), even if we could see a four-dimensional figure, our human vision is itself always two-dimensional, not three. (1) We can’t see entities in the fourth dimension, so how could there be a “visual image” of such a thing? Suppose we were to read this as a model of the 3D visual image formed as we look directly at a 4D hypercube? I propose that our figure is indeed a model, only not of the hypercube per se. What, then, can it be? It must be a “model” in some other sense. There simply isn’t elbowroom in our entire universe for the small object in the photograph! ![]() But the verdict is the same: there is no room for this figure in a space of a mere three dimensions. 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.Ĭounting 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. Yet I think we must be careful about the term model. It reveals the same fourfold symmetry, inherited as we saw from an origin in the cube. Its resemblance to the hypercube, albeit with its roof removed and a few important inner parts omitted, seems apparent. ![]()
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