Colours & Maps
In a recent newsletter we were asked, “How many ways are there to colour a cube (tetrahedron, octahedron,..) using each colour once?” If you tackled the problem, you’ll soon have realised that it isn’t the (simpler) one of colouring a row of boxes:
Here, there are 6 choices for the first box, 5 for the second, 4 for the third,… so that the number of ways is equal to 6 x 5 x 4 x 3 x 2 x 1 (“factorial” 6, written “6!”). The symmetry of the shape means we have, as it were, to lock it in position before we start colouring. For example, if we colour opposite faces of the cube red and blue:
We can rotate it about the marked axis into 4 positions and it looks exactly the same. This means that, if we colour one of the remaining faces yellow, we might equally well have chosen any of the other 3 for this colour.
However, this freedom is not unrestricted. You may have found from your investigations that colouring a solid with n sides, using each colour once, is a multiple of (n -3)!
(Did the number you found for the tetrahedron divided by (4 - 3)! = 1! = 1? Yes, it did!
Did the number you found for the cube divide by (6 - 3)!= 3!=6?
Did the number you found for the octahedron divide by (8 - 3)! = 5! = 120?)
We were also asked to think about ways of colouring a solid where every face was not necessarily a different colour. A particularly interesting way is to “map-colour” it: I’ll explain what I mean by this.
You may have heard of the 4-colour map theorem. This says that no map, however complicated, needs more than 4 colours to ensure that countries touching are coloured differently. This doesn’t just apply to a flat map:
We can bend it round a sphere – it doesn’t matter that it gets crumpled – so that the country surrounding all the others (or it might be the ocean surrounding a continent) acquires a boundary.
Now, the “sphere” can be a polyhedron. And on this, each country can be a face. Our requirement becomes: no faces touching edge-to-edge may be the same colour. Because we know that we shall never need more than 4 colours we can build our solids from Polydron or Clixi since shapes in both kits are supplied in red, green, blue and yellow.
What about the regular solids: how many colours are needed for the tetrahedron? The cube? The octahedron? The colouring of the pentagonal dodecahedron is particularly beautiful: each colour is symmetrical about a rotation symmetry axis of order 3:
But there’s a lot more to discover. How many different colour assortments meet at edges and how many times does that kind of edge repeat? How many meet at corners (“vertices”) and how many times does that kind of vertex repeat?
What about solids where every vertex looks the same and every face is a regular polygon but they are of more than one kind (“semi-regular” solids)? What about solids made from rhombuses? What about..?
Once you’ve found the smallest number of colours needed for your shape, ask, “How many different ways are there to colour my shape in this way?” Also ask, “How do I know that this solid is going to need at least 3 colours while that one may only need 2?”
In the same newsletter we were looking at tessellations. This is one of the 3 “regular” ones and doesn’t look very interesting…
But imagine that you map-colour a regular tetrahedron with wet paint and sit it on one of those triangles. You then roll it over edges so that it lands on adjacent triangles and colours them. What results? Build your tetrahedron in Polydron or Clixi. Roll it. Using the same kit, make the carpet it prints.
The tetrahedron is the only regular solid which prints a carpet without smudging. Prove that the others fail. Does/do any other solid/s print without smudging and cover the plane without gaps? If you think the answer is “Yes”, try to find it/them; if you think the answer is “No”, try to prove it.
Paul Stephenson
The Magic Mathworks Travelling Circus
