(1) We canвЂ™t see entities in the fourth dimension, so how could there be a вЂњvisual imageвЂќ 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 вЂњmodelвЂќ in some other sense. There simply isnвЂ™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 вЂњmodelвЂќ means a small-scale version of the real thing, this canвЂ™t be a model of the hypercube! This is evidently a photograph of some built thing вЂ“ and no four-dimensional entity can be squeezed into a three-dimensional space.Д¬ounting plane faces and allowing for the missing roof, this is a regular solid of twelve square faces вЂ“ 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.

Four dimensional hypercube