Hypercubes
Text and images by Nicholas Mee, author of Higgs Force.

The most famous example of a higher dimensional object is the hypercube. Broadly speaking, the hypercube is to the cube as the cube is to the square. But, what does this mean? What can we determine about the geometrical properties of the hypercube?

The best way to grasp a picture of the hypercube is to work up to it dimension by dimension. We can start in zero dimensions and proceed onwards to produce a whole family of cubes. A point does not have any extension. It is the unique zero dimensional object. If we allow our point to sweep out a length of one unit, say along the x axis, we can produce a line segment. Next we can allow the line segment to sweep out a distance of one unit in a perpendicular direction, say one unit along the y axis. The figure that we will produce is a square. Next, our square can sweep out a distance of one unit in a direction perpendicular to the first two directions, say the direction along the z axis. The result will be a unit cube.

Sweeping a square in the perpendicular direction generates a cube.

What happens next? We now seem to have run out of perpendicular directions in which we can continue this procedure. However, we can imagine that there are further perpendicular directions and create abstract higher dimensional spaces in our minds. There is no reason why we should restrict ourselves to the three dimensions of physical space when considering mathematical objects. As long as we reason clearly and consistently, we can deduce the mathematical properties of higher dimensional objects, even if they cannot be physically realised in our three-dimensional universe. We can simply declare that we are now working in four-dimensional space and see where our exploration leads us. If we find ourselves in an interesting place, then that will be sufficient reward for our journey. Often the research of pure mathematicians is guided by the search for interesting abstract scenery rather than a quest for scientific utility. In our abstract four-dimensional space, there will be a fourth axis that is perpendicular to our other three axes. In order to give it a name, we can label this axis the w axis. Measuring the distance parallel to each of the four axes gives us four coordinates that will specify the position of any point within our four-dimensional space.

With the fourth perpendicular direction that we have labelled the w axis, we can now take the cube and sweep it one unit in this direction. The object that we will generate is the four dimensional equivalent of the cube. It is the object that we are seeking, the hypercube. So what does it look like? Is there any way that we can visualize it?

Our sections through a cube formed a sequence of squares. And similarly, our sections through the hypercube form a sequence of cubes. What else can we infer from this description of a hypercube? If we draw the hypercube with an exaggerated perspective it will help us to understand the analogy with a cube. First we can draw a transparent cube face-on, so that we can see the back face within the front face and with edges connecting the corners of the front face to the corners of the back face. We can make a similar drawing of a transparent hypercube. If we position the hypercube in the same orientation as in our previous discussion, in our perspective drawing we will see a large cube, with a smaller cube within. Each of the corners of the outer cube are connected by edges to the corners of the inner cube. What we are looking at is the front cube (the outer cube) with the back cube, which is further away in four-dimensional space, appearing within it because its size appears diminished with distance. This is completely analogous to seeing the back face of the transparent cube within its front face.

In the projection shown above, the front and back squares of the cube (on the left) both look square, but the other four square faces of the cube do not look square. Similarly, in the projection of the hypercube, the front and back cubes of the hypercube both look cubical, but the projection has distorted the shapes of the other six cubes that form the hypercube. However, there are more symmetrical ways to project a hypercube down to two dimensions, in which the distortion is shared equally between all eight cubes. Such a projection is shown below. It gives a better feel for the structure of the hypercube. The outlines of each of the eight cubes can be picked out in this projection.

In the illustration shown above right, the axes have been given arbitrary labels: 'w', 'x', 'y' and 'z'. In four-dimensional space these would be the four mutually perpendicular axes. In this projection down to two dimensions, the 'w' and 'y' axes remain perpendicular and the 'x' and 'z' axes remain perpendicular, but the angle between the 'w' axis and the 'x' axis is now 45.

On the left above, the twelve edges of one of the 8 cubic cells of the hypercube have been coloured red. The edges and faces of this cube are aligned with the 'w', 'x' and 'z' axes. In four-dimensional space, this cube is perpendicular to the 'y' axis. On the right, the edges of the opposite cubic cell have been coloured magenta. If we were looking along the 'y' axis, these would correspond to the front and back cube of the hypercube. Note that each corner of one cube is connected to a corner of the opposite cube by a blue edge that is oriented in the 'y' direction, the direction that is perpendicular to the two cubes.

The image below shows a three-dimensional projection of the edges of a hypercube. The shadow of the hypercube is the two-dimensional projection that we have been considering above. In a sense the three-dimensional figure is a three-dimensional shadow of the four-dimensional hypercube.

Click the image to view an animation of the hypercube.

The Net of a Hypercube
Text and images by Nicholas Mee

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.

Albrecht Durer
self portrait (1500)

The net and two projections of an icosahedron

icosahedron

icosahedron

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.

cubenet2

The net of a cube

sqnet

Folding the net of a cube

The net of a hypercube

hypericon2

Folding the net of a hypercube

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 of 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.

cubeframe hyperframe
hypnet

Folding the Hypercube Net by Nicholas Mee. Click the image to view the animation.

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 - Crucifixion (Corpus_Hypercubus) by Salvador Dali]

To find out more about higher dimensional objects, click here.

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Copyright © 2010 Nicholas Mee. All Rights Reserved.