Where is the fourth dimension hidden?
A net of the hypercube allows us to represent the hypercube in three-dimensional space, but it is not the hypercube. We can wonder where the fourth dimension is “hidden” and how it is possible to make up – with some cubes – a four-dimensional object like a hypercube. In fact, the eight cubes, that we see, represent the exterior of the hypercube; yet, the fourth dimension “lies” in the interior of the hypercube……
To understand and to make this sentence more precise, let us go back to the examples we know. For a square, we have segments which are one-dimensional. They can be arranged in the plane in order to delimit a finite region of the plane: a square. For a cube, we have squares which are two-dimensional. They can be glued edge by edge in three-dimensional space in order to delimit a finite region: a cube.
In both cases, the dimension we “obtain” corresponds to the region we are delimiting by making up our objects, that is to say, the region in the “box” we are building. In the same way, the cubes of a hypercube can be glued in four dimensional space and the region delimited by the cubes is four-dimensional, one more than that of the cubes we begin with.
