The Net of a Hypercube

The six faces of a three dimensional cube can be unfolded into a crucifix shape that is known as a net of the cube.

There are 360° in a complete rotation. This means that we must rotate through an angle of 360° to orbit any point on a flat surface. If we cover a flat surface with polygons, then the sum of the angles that meet at any vertex must add to 360°. For instance, we can cover the plane with a grid of squares, with four squares meet at each vertex, because the angles of a square are 90° and 4x90° = 360°, as illustrated above.

In a polyhedron, such as a cube, the sum of the angles meeting at a vertex must be less than 360°, otherwise it would not be possible to fold up its net into the polyhedron. In a cube, three 90° angles meet at each vertex.

The great German Renaissance artist Albrecht Durer wrote a treatise on geometry and perspective in four books known as 'Underweysung der Messung mit dem Zirckel und Richtscheyt', which in English means 'Course in the Art of Measurement with Compass and Ruler'. The fourth book is about polyhedra and includes drawings of the nets of various polyhedra including the net of an icosahedron shown below.

Icosahedron means 'twenty faces' and Durer's net of an icosahedron contains twenty equilateral triangles, as can be seen from the illustration. Each of the angles of an equilateral triangle is 60°. When the net is folded up to form an icosahedron, five such angles are brought together at each vertex. The sum of the angles around each vertex of an icosahedron is therefore 300°.

In the diagram of the net of a cube on the left below, each pair of edges that must be glued together to form a cube are shown in the same colour. The net is two dimensional, a third dimension is required to fold the net and form the cube.

Similarly, we can construct the net of a hypercube. It is formed of 8 cubes. To construct the net, we take one cube and affix another cube to each of its six faces, then attach the eighth cube to the opposite face of one of these six cubes, as shown below. In the illustration, each pair of faces that must be glued together to form the hypercube are shown in the same colour.

To fold the net into a hypercube we require an extra spatial dimension. In three dimensions, we can fold the net of a cube around and join up its edges to enclose a region of three dimensional space. Similarly, in four dimensional space we could fold the net of the hypercube and glue the appropriate faces together to produce a hypercube.

If we look more closely at how the cube's net folds up on itself, we can extrapolate up a dimension and it will give us another insight into the structure of a hypercube. The first thing to note is that whereas a cube is glued along its edges, it is the faces that are glued together to form a hypercube. Also, whereas in a cube there are three square faces surrounding each the corners or vertices, in a hypercube there are three cubes around each edge. We can fill three dimensional space by stacking cubes, just as we can cover two dimensional space with a grid of squares. If we pack three dimensional space in this way, there will be four cubes around each edge. This is because the angle between two faces of a cube is 90°. But, in three dimensional space there is no way to glue three cubes around an edge without leaving a gap, so we can only fold up the net to form the hypercube in four dimensional space.

The hypercube bears the same relationship to the stacking of cubes to fill three dimensional space, as the cube bears to the two dimensional grid of squares.

In the net of a cube each pair of faces that are separated by an intermediate face will become opposite faces of the cube when the net is folded up. These pairs of faces are shown in the same colour in the net above. In the net of the hypercube, pairs of cubes that are separated by an intermediate cube will become opposite cubes, when the net is folded into a hypercube in four dimensional space. These pairs of cubes are shown in the same colour in the image of the hypercube net above. Each of these pairs will become the front and back cubes of the hypercube when it is viewed in the direction perpendicular to these cubes. This means that when the net has been folded to form the hypercube, the relationship between the cubes will be the same as the relationship between the inner and outer cubes as shown in the projection below.

Salvador Dali alludes to the transcendental nature of four dimensional space in his famous painting 'Crucifixion - Corpus Hypercubicus' (1954), which depicts a vision of Christ crucified on a cross that is formed from the net of a hypercube. The following link will take you to a reproduction of Dali’s painting in Wikipedia.

[Illustration - http://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus)]