When we point to a picture like this and say that it is a hypercube, we are leaving out something important. To discover the properties of a hypercube, we must clarify what it is that we have left out. In fact, the picture is not that of a hypercube, which is a four-dimensional object, but that of a net of a hypercube, which is a three-dimensional object (and the picture, of course, is only a two-dimensional representation of that three-dimensional object). This three-dimensional net is related to the real four-dimensional object in the same way that this plane surface consisting of six squares is related to a cube.
The plane surface becomes a cube when the faces are folded out of the plane and the boundary edges are identified two by two. In a similar way, we can decide which edges of the net are to be identified.

The eight cubes of the net fold into a hypercube when certain boundary faces are identified and glued together two by two. This time too, we can decide which squares in the net may be identified.
Of course, not all arrangements of eight cubes (octacubes) are nets of a hypercube (just as not all figures composed of six squares form the net of a cube). For instance, this is not a net of a hypercube. The problem is that there are there four cubes adjacent to the same edge. It is similar to the analogous situation of placing four squares adjacent to a vertex in the plane in trying to form the net of a cube. Even if we could leave the plane and move into three-dimensional space, we could not fold the figure into a cube because we would be prevented from doing so by that vertex.

There are 11 nets of a cube altogether and there are 261 nets of a hypercube. This video shows some of these 261 nets. They are obtained from one another by allowing the cubes to roll along the octacube in analogy to how we could obtain the 11 nets of a cube by removing one square and gluing it somewhere else (at a valid place, of course). An interactive animation shows all the nets. If you click on a face of a net, the octacube changes; a part comes off and is glued back along the square that has been clicked. Note that you should not click on just any face, but only on a square that is adjacent to an edge where three cubes come together. You can detect the clickable squares by the fact that they light up when they are selected. Another animation proposes to compare a fixed net and other five ones: you must recognize the fixed one among the others. It is not at all a trivial task!

How do we know that there are 11 nets of a cube and 261 nets of a hypercube? And in general, how can we discover all the nets of a given polyhedron or polytope (a polytope is the generalization of a three-dimensional polyhedron to other dimensions)? The answer to the first question is easy. You can obtain them “by hand” by examining all possible figures made up of six squares. The answer to the second question is more complicated, but in principle can be worked out in the same way as the first question. The answer to the third question is more surprising even in three-dimensional space, never mind bothering about four dimensions. If you enjoyed finding out that there are 11 nets of a cube, you can easily discover the two nets of a regular tetrahedron, and with a little effort, the eleven nets of a regular octahedron . On the other hand, you will be in for a lot of work if you try to count all possible nets of a dodecahedron or a icosahedron by hand. In both cases, there are 43380 of them.

Let us imagine to be an ant on a cube. The ant is in the middle of one of the six square faces. It moves toward the center of one of the four adjacent faces and, after that, it goes “straight ahead”. “Straight ahead” is in inverted commas because the ant does not move on a straight line. It must move on the surface of the cube, so sometimes there are right angles on its path.
It is easy to realize that the ant goes back to the starting point after moving on four squares which form a sort of a “ring”. The four squares are all the faces of the cube except two, which are opposite one to the other.
On a cube there are three different rings like the one mentioned before.
What happens on a hypercube? What kind of path will a hyper-ant describe if it starts from the center of one of the eight cubic faces and goes “straight ahead” aiming at the center of one of the six adjacent cubes?

The path described by the hyper-ant closes up this time as well. The hyper-ant goes back to the starting point after passing on four cubes. The chain of four cubes can be visualized quite well on a net of the hypercube. In particular, it is clearer on a net where there are four cubes one over the other. When we make all the identifications and pass to the four-dimensional world, we must recall that, on the column of four cubes, the bottom face of the bottom cube is glued to the upper face of the upper cube. In this way, the four cubes form a doughnut. Surprisingly, the cubes “avoided” by the ant form a doughnut too, which is linked to the previous one. This can be well visualized in some nets of the hypercube. This means that the hypercube splits in two linked doughnuts, each of four cubes.

How many doughnuts of this kind are there in a hypercube? We can start from a cube and point out that there is a doughnut - which contains the cube - for each of the three directions orthogonal to two opposite faces in the cube. Since there are 8 cubes, we obtain 24= 8 × 3 doughnuts. Each of these doughnuts is counted 4 times, one for each of the cubes in it. In other words, there are six different doughnuts in a hypercube.

In the animation about the hyper-ant you can see the six different possible paths, that is to say, the six possible doughnuts of four cubes in a hypercube (and the three possible decomposition of a hypercube in two pairs of doughnuts). On the left of the screen, the “environment” of the hypercube is described under stereographic projection.The hypercube is enflated on a hypersphere which is next projected on three-dimensional space. On the right of the screen, you can follow the same paths on a net of the hypercube. The colours are not randomly chosen. They highlight the directions of the two-faces. The 24 squares in the hypercube are coloured with six different colours; for each colour there are 4 (parallel) faces which are coloured with that colour. The six possible paths of the hyper-ant correspond to the six different colours, i.e., each of them is “orthogonal” to the 4 squares of a fixed colour.

Some virtual leaflets ("The nets of a hypercube" and "How to split a hypercube in two doughnuts") go over the structure of a hypercube.

In others you may find some different shapes of four dimensional world. In "Why only six?" there are the analogue of the regular polyhedra and in "Not only hypercubes" you may find the analogue of the sphere and of the star polyhedra. Finally, in "Beyond four dimensions" you may see other constructions which can be repeated in higher dimensional space.

As we can imagine a hypercube or other objects in four-dimensional space, could we think of 5, 6, or more, dimensions? The answer is positive and it is not even more difficult conceptually: at the most it is our imagination to be challenged. For instance, we can think of the n-dimensional space as the set of n-tuples of real numbers. As for the plane, we can introduce a coordinate system and identify a point on the plane with a pair of real numbers; in three-dimensional space any point is identified with a triple of numbers. Therefore, an n-dimensional space has the right to exist for any n. Of course, this does not help much to imagine objects in this space.

Let us go back to a hypercube. We obtain a hypercube from a cube with the same abstract procedure as the one we use in order to obtain a cube from a square. This will be called the prism construction. According to it, we can start from a point (dim. 0) and imagine to obtain a segment (dim. 1); next a square (dim. 2), next a cube (dim. 3) and finally a hypercube (dim. 4).
There is no reason why we should stop: the prism over a hypercube of dimension four will be a hypercube of dimension 5 and so on.
Also using coordinates, we can center our objects in the origin and arrange them so that the edges are parallel to the coordinate axes. Then a square (with edge two) has four vertices (±1,±1); The vertices of a cube are the eight points (±1,±1,±1); The vertices of a hypercube are the 16 vertices (±1,±1,±1,±1). So forth, we will obtain that the vertices of a hypercube in n-dimensional space are the 2n points (±1,±1,...,±1,±1).

There are other constructions which can be repeated infinitely many times, for instance, the pyramid construction. It can be applied to obtain a tetrahedron from a triangle. It suffices to join a point not on the plane where the triangle lies with all the vertices of the triangle. If you start with an equilateral triangle you will get a regular tetrahedron if the following conditions are satisfied: i) the point is on the line perpendicular to the plane where the triangle lies and passing through the barycenter, and ii) the height is taken in such a way that all faces are equilateral triangles.
Coordinatewise, the easiest way to visualize this construction is to start from a triangle on the plane x+y+z=1 in three-dimensional space. The vertices are the points (1,0,0), (0,1,0), (0,0,1).
In (n+1)-dimensional space, the n+1 points (1,0,...,0),...,(0,...0,1) are vertices of a hypertetrahedron in dimension n.

Another construction, that we will call the bipyramid, allows one to get a regular octahedron from a square and, at the next step, a hyperoctahedron in four dimensional space from an octahedron.

Coordinatewise, the four points (±1,0) e (0,±1) are vertices of a square. The six points (±1,0,0), (0,±1,0), (0,0,±1) are vertices of a regular octahedron. In general, the 2n points (±1,0,...,0),..., (0,...,0,±1) are vertices of a hyperoctahedron in n-dimensional space.

It is interesting to note that in dimension greater than or equal to 5 one can prove that these are the only regular polytopes. The 120-cell, the 600-cell and the 24-cell are specific “accidents” of dimension four. Analogously the icosahedron and the dodecahedron are specific “accidents” of three-dimensional space.

In these pages there are several pictures, animations and information on hypercubes. One might think that hypercubes are the only objects in four-dimensional space.....: it is not true! It is only the simplest object and, therefore, it is easier to describe it. In four-dimensional space there is a "zoo" of polytopes which is richer than the "zoo" in three-dimensional space.

To begin with, there are six regular polytopes, which are the analogue of the five regular polyhedra in three-dimensional space: one of them is the hypercube.
There are other possibilities as well. One can define and investigate various families and objects in the 4D world as in the 3D world exist polyhedra , prisms and pyramids, uniform polyhedra and star polyhedra.

Here you find one of the animations on a regular star polytope. Precisely, it is the analogue of the great icosahedron . In the animation this polytope goes through a 3D world. We see a section of it as if an inhabitant of flatland saw all the different polygons, which take shape in his/her flat world, when a star icosahedron goes through it.

It is not surprising that one sees several “detached” polyhedra at the beginning. They look like the great star icosahedron. If we imagine taking a section of the great star icosahedron with a plane which is orthogonal to the direction vertex-vertex, one would see a regular star with five points. If we cut it with a plane which is orthogonal to the direction face-face, one would see three "detached" stars at the beginning.

They would not be regular stars but they would appear a bit deformed.

What else can we find in four-dimensional world? Can we find the analogue of surfaces? Spheres, for instance? Of course, we can. Furthermore, the analogue of surfaces yields a variety of shapes, which is richer than the relative monotony of surfaces.
Let us talk here about the sphere. The hypersphere in four-dimensional world can be defined like the sphere in 3D (and the circle on the plane), that is to say, like the set of points with the same distance from a given point, which is the center of the hypersphere. How can we imagine it? In a sense, our world is a hypersphere as long as we "close it up" with a point at infinity.

To explain this claim, suppose you put a sphere on a plane and next project the sphere on the plane from the North pole (the antipodal point with respect to the tangency point betweeen sphere and plane). We obtain a bijective correspondence - the so-called stereographic projection - between the whole plane and the sphere except the North Pole. Analogously, there exists a correspondence between the 3D space and a hypersphere with a point removed - we still call it stereographic projection.

We can project onto the 3D space (and so "see") some objects contained in a hypersphere. This is the principle on which the following animations are based. In this one, imagine to project the 120-cell on the circumscribed hypersphere (as you can “enflate” a dodecahedron on the circumscribed sphere . Via the stereographic projection, one can pass from a hypersphere to 3D space and imagine to "get into" a 120-cell.

Of course, the faces are not regular dodecahedra, just as the faces of the dodecahedron, which is here drawn, are not exactly pentagons; these are the drawbacks of projections....!

If we want to make a regular polyhedron, we start up with regular polygons that are all alike. We assemble them around a vertex so that there is a gap, in the plane. In other words, the sum of the angles around the vertex is less than 360°.

It is exactly this gap that allows closing up the polygons in three-dimensional space; so we obtain a convex polyhedron. If the sum of the angles were equal to 360°, we would obtain a plane tasselation).

That's the reason why the regular polyhedra can not be more than five. Indeed, the possible combinations are five: 3 by 3 pentagons, 3 by 3 squares, 3 by 3 triangles 4 by 4 triangles or 5 by 5 triangles.
Clearly, in order to be sure that the regular polyhedra are exactly five, we should prove that each combination around a vertex can be carried out with the same pattern so to obtain a polyhedron.
Let us try to mimic this argument in higher dimension, i.e., let us try to make a polytope in four-dimensional space. We can start to choose a regular polyhedron and then try to assemble various copies of this polyhedron, all alike, around an edge. As before, we need a gap (this time in three-dimensional space). In other words, the sum of the dihedral angles at an edge must be less than 360°; As for the polygons that come up in a polyhedron, this gap will be responsible for closing up the polyhedra in four-dimensional space; so we will get a convex polytope. If the sum of the dihedral angles were equal to 360°, we would obtain a space tessellation. For example, think of four cubes around an edge.

Also in this case there are not too many possibilities. If we start with regular tetrahedra, we can assemble them 3 by 3 or 4 by 4, or 5 by 5, and not more. In fact, the dihedral angle of a tetrahedron is greater than 60° and smaller than 72°.

If we start with cubes, we can assemble them 3 by 3 and not more because the dihedral angle of a cube is 90 degrees. If we use four cubes, we will tessellate the space.
If we start with regular octahedra, we can assemble them only 3 by 3 because the dihedral angle of a regular octahedron is greater than 90°(and smaller than 120°).
If we start with regular dodecahedra, we can assemble them only 3 by 3 because the dihedral angle of a regular dodecahedron is greater than 90° (and smaller than 120°).

If we start with regular icosahedra, we can not assemble them even three by three because the dihedral angle of a regular icosahedron is greater than 120 degrees. There are at most six possibilities. In fact, one can prove that each of these combinations can be repeated with the same pattern so to obtain (in four dimensional space) a convex polytope:

• the hypertetrahedron which is made up of 5 tetrahedra (three at each edge);
• the hyperoctahedron which is made up of 16 tetrahedra (four at each edge);
• the 600-cell which is made up of 600 tetrahedra (five at each edge);
• the hypercube which is made up of 8 cubes (three at each edge);
• the 24-cell which is made up of 24 octahedra (three at each edge);
• the 120-cell which is made up of 120 dodecahedra (three at each edge).