If we unfold a hypercube, should the cubes not disappear upon making the identifications?

The six squares in a net of the cube are the six faces of the cube and the eight cubes in a net of the hypercube represent the “cubic faces” of the hypercube. In other words, the eight cubes represent the “exterior” of the hypercube, that is to say, what you should “see” of the hypercube in four-dimensional space.

On the other hand, if we imagine to glue the faces of the cubes two by two (as required by a net) , the faces should not be visible any more and not even the cubes! But if you can not see the eight cubes, where is the hypercube?

This time, the analogy is extremely difficult since we must study not only the three-dimensional analogue, but we must also suppose that we are “two-dimensional” observers. More specifically, we must imagine that we are two-dimensional beings who live on a plane where you find two squares, each of them with three black edges and one red edge.

Now, let us suppose that the two squares are glued red edge by red edge. We should see the exterior of a rectangle with four black edges; the red edge would disappear from our sight! But if you raise your head from the plane, you can still see the red edge.

Likewise, if two cubes have a face in common in four-dimensional space, you can still see this face (it is “external”). In particular, you can still see each face of the eight cubes in four-dimensional space although they are glued face by face.

In the following virtual leaflets you may find some FAQ on the exhibition Diving into the fourth dimension, which was held during the Festival della Scienza in Genova, 2007.

In each representation of a hypercube you should “see” each of the eight cubic faces of the hypercube. This is clear for a net, but what about a Schlegel diagram and a projection? Where are the eight cubes?

In a Schlegel diagram, you can see that seven out of the eight cubic faces of a hypercube are represented. A small cube inside and other six deformed cubes - which are glued face to face - and form the big cube outside. Where is the eighth cube? The faces of the eighth cube are the faces of the big cube. Since the interior of the big cube is already “taken” by the other seven cubes, we can imagine that the interior of the eighth cube is the exterior of the big cube, which is an unlimited region!
This remark comes from our way of thinking in three dimensional space. The Schlegel diagram of the cube in the plane can help us understand the hypercube.
We notice that also in a Schlegel diagram of a cube a face is missing. It is the face delimited by the big square. Actually, this is a two-dimensional projection. In fact, we should realize that the other five faces are in close-up (like a relief) with respect to the sixth face, the one delimited exactly by the edges of the big square.
Likewise, in the diagram of a hypercube the seven cubes, which are inside, must be imagined in “close-up” with respect to the cubic face delimited by the big cube. This perspective is less evident than in the case of a cube because we are adding a dimension to our three-dimensional space to highlight what is in the big cube; but this is “watching” in a four-dimensional space…
As for a perspective projection of a hypercube two cubes are easily detectable. Where are the others? Let us consider a projection of a cube on the plane. The projection of the six faces are intertwined. In the same vein, the projection of the eight cubic faces of a hypercube intersect each other. To observe them, let us consider the bases of these two cubes.

You can easily see that they are the bases of two other solids (first, second) with eight vertices and twelve edges, which are the projection of two other cubic faces.
There are still four cubes missing: we leave it to the reader to discover where they are “hidden”……

Why do we need eight cubes for a net of a hypercube?
To understand it, let us examine the cases of a square and a cube:

• For a square each edge is adjacent to other two. The two vertices of these two edges, which are not in common with the original one, are the vertices of the edge opposite to the original one. The number of the edges is thus 1+2+1=4 (the first one, the two adjacent ones and the opposite one);
• For a cube each square is adjacent to other four squares (it has four edges); the “free” edges of these four squares are the edges of the square opposite to the original one. The number of the faces is thus 1+4+1=6 (the original face, the four adjacent ones and the opposite one).

Both in the case of a square and a cube, we have the following situation: each ”face” (an
edge or a square respectively) is adjacent to “almost” all the other faces, that is all except one of them.

Let us try to proceed by analogy for a hypercube. Since each “face” of a hypercube is a cube, with six faces, it is reasonable to think that there are 1+6+1=8 cubic faces of a hypercube (a cube, the six adjacent cubes and the opposite cube). We are led to assume that a hypercube has eight cubic faces.

How many cubes are there really in a hypercube? Infinitely many!


To understand this statement, let us consider again a cube and proceed by analogy.

A cube has six faces, which are identical. Actually, there are infinitely many squares in a cube. For example, let us consider a cube and a plane parallel to a face. If we imagine to let the cube pass through the plane, the intersection of the plane and the cube is a square at every time.

MANCA UN PEZZO!!!
Allo stesso modo immaginate un ipercubo che attraversi un universo tridimensionale come il
nostro, che sia parallelo a una delle sue otto facce cubiche: l’intersezione
dell’universo 3d e dell’ipercubo è un cubo e quindi ad ogni istante apparirebbe come un cubo.

A cura di Paolo Bellingeri per la mostra Un tuffo nella quarta dimensione.

A net, a Schlegel diagram or a perspective projection: which of these three methods best represents a hypercube? The answer is: “ It depends”. In fact, they are three-dimensional representations of a hypercube, but they are not the hypercube. A representation may be more fit to study a property than the others.

For example, a Schlegel diagram seems to be the best for some combinatorial problems. On the other hand, it is more appropriate to choose a net for some optimization problems, such as finding the shortest path between two points. And yet, a projection shows more clearly that a hypercube contains infinitely many cubes. Having more than one representation allows one to get more information on the hypercube and its properties.

In other words, we do not take into account the hypercube, but we study an (easier) picture of it from which we infer some properties of the hypercube.

Likewise, mathematicians have defined various “representations” of four-dimensional space, which are very diverse (algebraic, analytic….). They are more or less complicated and difficult to work with, but they allow mathematicians to observe and describe various characteristics of four-dimensional space.

As recalled before, a net, a Schlegel diagram and a projection are three-dimensional objects; however, it is worth mentioning that a net is the only one that differs from the other two because a Schlegel diagram is essentially a projection. If we identify the faces of a hypercube as required, we get a real hypercube. Actually, we can not do it in practice because our space is three-dimensional. It is a widely common technique in Topology to make up objects of higher dimension via identification of faces, segments, points, namely objects of smaller dimension. This technique allows one to make a great deal of diverse objects from simpler shapes.

For example, consider a square and identify the two pairs of opposite edges. If you assume the square be flexible and deformable, you can make these identifications in three-dimensional space: you will get a doughnut.

If you give a half-twist in one of the two identifications before you glue a pair of opposite edges (so as to change their direction), not only is the doughnut unobtainable, but it is actually not feasible to deform a square in three-dimensional space without forming any self-intersections. On the other hand, it is possible to make these identifications in four-dimensional space. This object is known as the Klein bottle.

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……