CUBE {4, 3}

Simple cube properties

The cube is the most commonly occurring of the regular polyhedra, and is in many ways the simplest in its structure.

The cube is sometimes called a hexahedron, from the Greek hexa + hedron meaning six bases (faces). The basic properties of the cube are as listed below.

 Name Symbol V E F V – E + F Cube {4, 3} 8 12 6 2

Each face is a square, each vertex figure an equilateral triangle. If the cube has edge-length s, then the volume is s3 and the surface area 6s2. The face angles and dihedral angles are clearly all 90°. The cube has an obvious centre where the long diagonals meet. This point is the centre of gravity, incentre, midcentre and circumcentre.

At right is the Schlegel diagram for the cube, giving an alternative verification for Euler’s formula. Can you see how this represents the cube?

(a) It is simple to calculate the inradius, midradius and circumradius of a cube of sidelength 2s!
(b) How many long diagonals does the cube have? What is their length? the angle between them? Are you sure?

A cube of side length 1 is often called the unit cube.

Here is a summary of some of the metric attributes of a cube of side length s.

Model making

It is easy to make your own model cube. A net and face template of suitable size is given here. For experimenting with a number of cubes, it is possible to buy sets of small plastic cubes.

A basic net (without tabs) for constructing the cube is shown at right. However this net is by no means unique – there are in fact eleven distinct nets possible. Notice that the squares of any net must be edge-connected, and we do not want overlapping faces. Also, two nets where one can be obtained from the other by reflection or rotation are regarded as being the same.

Find as many of the eleven nets as you can, making a sketch of each.
Can you suggest any strategies here? For example, how many squares do you expect your net to contain? Does any net have to ‘span’ at least three squares in the horizontal and vertical directions? Are there any configurations of squares which are disallowed?

Find as many of the eleven nets as you can, making a sketch of each.
Can you suggest any strategies here? For example, how many squares do you expect your net to contain? Does any net have to ‘span’ at least three squares in the horizontal and vertical directions? Are there any configurations of squares which are disallowed?

The table below contains a number of suggested nets. Click on the suggestions which you think are correct.

Obviously each net must have six squares, corresponding to the six cube faces. All but one of the nets span at least three squares horizontally and vertically: the odd one is a bit of a surprise! Four squares about a point is one of the disallowed configurations, obviously leading to an overlap when folded.

Space tessellations

Remind yourself of the three regular plane tessellations we investigated earlier. What properties did we require?

Do you think a regular tessellation of space might exist? Would the cube be a suitable ‘tile’?

In fact the cube is the only Platonic solid which will tessellate space. Surprisingly, perhaps, the regular tetrahedron does not have this property. However, there are a couple of non-regular polyhedra which can easily be seen to tessellate space.

If we look at the cube ‘face on’, we see the square which is a regular polygonal tile for the plane. Perhaps we can replace this polygon (and the opposite cube face) with another regular polygon, to obtain a ‘semiregular’ polyhedron which will tessellate space? Think about this; it is a good mental exercise to visualize shapes in space.

If we look at the cube ‘face on’, we see the square which is a regular polygonal tile for the plane. Perhaps we can replace this polygon (and the opposite cube face) with another regular polygon, to obtain a ‘semiregular’ polyhedron which will tessellate space? Think about this; it is a good mental exercise to visualize shapes in space.

Each of the planar regular tessellating tiles can be used as the defining face of a prism (with all faces regular polygons) which will tessellate space. Of these, only the cube involves just one type of polygon, so (of these) only the cubic tessellation is regular. We leave as an open question for now whether there are any other interesting ‘semiregular’ tessellations of space.

 Further properties Although the cube is such a simple and well-known figure, it still has some surprising properties. We will investigate some of these using our applet. Choose the cube setting in the menu. Remember that the (parallel) projection of a solid can be thought of as the shadow it casts under exposure to parallel light rays. •  Play with the applet to obtain the following sequences of projections. (a) square, rectangle, square. What are the dimensions of the (maximum) rectangle? Where have you seen this shape before? (b) square, regular hexagon, rectangle, regular hexagon, square.

The maximal rectangle which appears in the problem above has edges in the ration 1 : 2. This is the shape of the commonly used A3 and A4 sheets of paper. This rectangle has the property that if you cut it in half across, the two resulting smaller rectangles have the same shape. This is obviously ideal for paper manufacture.

We observe that   2 : 1 = 1 : 2/2.

We next explore some rather different properties.

(a) In the applet, hold the cube still, with one face closest to you. Suppose now that you take a sequence of slices (or sections) through the cube, parallel to, and starting from the near face, and moving towards the far face. What shape are they? (Trivial!)

(b) Next, hold the cube with one edge closest to you, and the opposite (not visible) edge furthest away. Now take sections parallel to ‘the plane of the screen’ starting from the nearest edge, and working towards the farthest edge. What shapes do you get? What do they have in common? How do they differ? (pretty trivial too!)

(c) Finally, hold the cube with one vertex closest to you, and again take sections parallel to ‘the plane of the screen’ starting from the nearest vertex and working towards the farthest vertex. If you have not thought about this before, you may find this exercise rather difficult. What shapes do you get? And what happens in the middle? Write down your answers. These are quite unexpected shapes for a cube section.

Finally, and just for fun

Experiment with the different applet views to obtain
(a) A hexagon subdivided into three congruent rhombi (what angles appear here?)
(b) A hexagon subdivided into six equilateral triangles.

Thinking about the sections, we clearly obtain squares and a sequence of rectangles in the first two cases. But it is (c) which is really interesting. The behaviour can be seen here ... .

The triangles which have diagonals of the cube as their edges correspond to faces of the inscribed tetrahedron which we observed earlier.

Two cube problems

A very curious problem asks whether it is possible to make a square-shaped hole through a given cube, leaving the cube ‘all in one piece’, and such that a larger cube can be passed through it. Surprisingly the answer is ‘Yes’: it is possible to pass a cube of side length 32/4 = 1.06065 through a cube of side length 1. The cube with its square cross-sectioned channel is shown at right.

The largest cube involved in this problem is generally known as Prince Rupert's cube. But who was Prince Rupert? The link is uncertain, but a Prince Rupert was born in Prague, Bohemia in 1619, and became the most talented commander of the English Civil War (1642 – 1651). In the years before his death in 1682, he dabbled in scientific experiments, making him a likely candidate.

Here is another cube problem called ‘box in a box’.

• Suppose you are given a cube of side-length 1. What is the dimension of the smallest cube that can be inscribed in this cube, such that the inscribed cube makes contact with each face of the original cube?

To see the solution, check here. It would be nice to be able to provide a simple proof for this rather nice result. Notice that the contact points are not the centres of the faces of the outer cube.

In correspondence about this, David Eppstein writes: ‘The only solutions I have been able to find involve the smaller and larger boxes sharing a common long diagonal; the best of these solutions orients the smaller box 180° rotated around that diagonal, with side length 3/5 that of the larger box. Six of the vertices of the smaller box are on the faces of the larger box, 3/5 of the way along (one of) the face diagonals; the remaining two vertices are 1/5 and 4/5 of the way along the long diagonal of the larger box.’

Vertex coordinates

As with the tetrahedron, it is useful to be able to give a set of coordinates for the vertices of a cube. This can be done in many ways, but we are looking for a set which is simple, and which hopefully demonstrates a high degree of symmetry.

Sketch a cube, and add a set of three mutually perpendicular axes. Which position and orientation of the axes seem most symmetric with respect to the cube? Suggest a possible set of coordinates for the vertices of the cube.

Sketch a cube, and add a set of three mutually perpendicular axes. Which position and orientation of the axes seem most symmetric with respect to the cube? Suggest a possible set of coordinates for the vertices of the cube.

If we are looking for symmetry in the coordinates of the vertices, we are likely to take the origin (the intersection of the axes) at the centre of the cube, and the axes themselves parallel to the edges of the cube. Suppose now that the edges of the cube have length 2. Then the eight coordinates can be taken as (1, 1, 1), where the signs are taken independently. This is a wonderfully symmetric set of coordinates – certainly much better than this possibility for a unit cube:

(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1).

With the symmetric set of coordinates we have chosen, the 8 vertices relate to the 3 coordinates for each vertex by the relationship  8 = 23.

Real life occurrences

You should be able to think of many real life occurrences of the cube. It is the most frequently occurring of the polyhedra. Why do you think this is? Make a list here:

Here are some ideas.

 Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 Example 9

References

Properties

Cube properties : http://mathworld.wolfram.com/Cube.html

Model making

The excellent model book: Wenninger, M. J., Polyhedron Models, Cambridge (1971).

Number of nets:

Buekenhout, F., Parker, M., ‘The number of nets of the regular convex polytopes in dimension 4’, Discrete Mathematics 186 (1998) 69 – 94.

Turney, P. D., ‘Unfolding the tesseract’, Journal of Recreational Mathematics 17 No. 1 (1984 –85) 1 – 16.

Space tessellations

Space filling polyhedron : http://mathworld.wolfram.com/Space-FillingPolyhedron.html

Further properties

International paper sizes : http://www.cl.cam.ac.uk/~mgk25/iso-paper.html

Two cube problems

Prince Rupert’s cube : http://mathworld.wolfram.com/PrinceRupertsCube.html

Cundy, H. M., Rollett, A. P., Mathematical Models, Oxford (1961)

 (a) The inradius is s, the midradius 2s, and the circumradius 3s. (b) I find it instinctive to say that the cube has three diagonals, all mutually perpendicular. Of course this is quite wrong. There are four such diagonals, they have length 3, and the angle between them is                                            arcsin(1/3) = 38.94°.