|
CUBE {4, 3}
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.
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.
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 : We observe that We next explore some rather different properties.
Finally, and just for fun
The triangles which have diagonals of the cube as their edges correspond to faces of the inscribed tetrahedron which we observed earlier.
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.
(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 = 23.
Here are some ideas.
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:
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
|
Simple cube properties
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 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.
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 
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 3
• 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?
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, 
4’, Discrete Mathematics 186 (1998) 69 – 94.
(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 |