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
Each face is a square, each vertex figure an equilateral triangle. If the cube has edge-length 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 cube of side length 1 is often called the unit cube.
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.
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. Remind yourself of the three regular plane tessellations we investigated earlier. What properties did we require?
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.
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.
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.
Finally, and just for fun
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’.
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.’ 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.
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 (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 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 = 2 Real life occurrences
Here are some ideas. References Properties Cube properties : http://mathworld.wolfram.com/Cube.html Model making The excellent model book: Wenninger, M. J., Number of nets:
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 Box in a box : http://www.ics.uci.edu/~eppstein/junkyard/rect.html |

(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 |