Tilings

I am still ploughing through the first unit of M336 'Groups and Geometry' and there have been a lot of new ideas to digest concerning the topic of tilings. Sometimes I find the definitions hard to fix in my mind and it is a bit like learning a new language. For example, I found adjacency and incidence bothersome and I keep having to go back to the definitions in order to understand what they mean.

A tiling can be divided into three different parts; the tiles themselves, the vertices and the edges. Each of these parts has a particular meaning. The vertices are points on the boundary where three or more tiles meet and the edges are points on the boundary which join vertices and are where two tiles meet. Now the edges don't have to be straight lines, they can be curved. Also, if the tiles are polygons, the tiles can have corners which are not necessarily vertices and sides that can be made up of more than one edge (and vice versa). All very confusing. Some of this confusion is removed by demanding that a tiling is edge-to-edge, i.e. that each side of the polygon corresponds to exactly one edge and vice versa.

Now adjacency is a relationship between the same parts of a tiling whereas incidence is a relationship between different parts of a tiling. Now here is the definition of these two concepts.

Two distinct tiles are adjacent if they share a common edge. Two distinct vertices are adjacent if they are joined by an edge. Two distinct edges are adjacent if they share a common vertex and bound a common tile.

The edges and vertices on the boundary of a tile T in a tiling are incident with T. Similarly, T is incident with the edges and vertices on its boundary. Also, if an edge E joins vertices V and W, then E is incident with V and W and these vertices are incident with E.

Another important definition is the degree of a tile or vertex. The degree of a tile in a tiling is the number of other tiles to which it is adjacent. The degree of a vertex in a tiling is the number of tiles (or alternatively the number of edges) with which it is incident.

Some examples of Archimedean tilings can be found here. An Archimedean tiling is an edge-to-edge tiling with regular polygons that is vertex-uniform. Only 11 such tilings can be constructed (allowing for one to be a reflection of the other). The first three are the regular tilings and the other 8 are semi-regular. Under each tile is a number. For example under the snub hexagonal tiling is the number 3⁴.6. In OU parlance this is the vertex type (3,3,3,3,6). If V is any vertex in a tiling, then the vertex type of V is given by listing the degrees of the tiles incident with V, starting with any one of them and proceeding in either direction in clockwise or anticlockwise order. So you can see how you have to understand all these definitions!

If you look at the snub hexagonal tiling you notice that all the vertices look the same. Choosing any vertex, then incident with it are four equilateral shaped tiles and one hexagonal tile. If we look at the equilateral triangle tile, then there are three tiles to which it is adjacent, that is its degree is 3 (since this is an edge-to-edge tiling there are as many adjacent tiles as there are sides of the equilateral triangle). If we look at the hexagonal tile then its degree is obviously six. Thus going round the vertex that we chose in either direction we end up with a vertex type of (3,3,3,3,6) (allowing for cyclic permutations of these numbers). A tiling is vertex-uniform if all the vertex types in the tiling are the same.

Not only can you define a vertex type but you can also define a tile type. It is similar in definition to a vertex type, but instead you list the degrees of the vertices incident with a tile in a tiling. The tile type of our snub hexagonal tiling is not uniform. The tile type for the hexagon is [5,5,5,5,5,5] (notice that square brackets are used) but the tile type for the equilateral triangle is [5,5,5].

My method for remembering how to find a vertex type or a tile type is that for vertex types it is tile-tile (i.e. you are counting tiles around tiles incident to a vertex) and for tile types it is vertex-vertex (i.e. you are counting vertices around vertices incident to a tile).

First books

Well, I have made a start on the course books for 'Groups and Geometry' (M336) and 'Number theory and mathematical logic' (M381). I began with M381 and the first book, which I have now gone through, 'Foundations' deals with four topics; Numbers from Patterns, Mathematical Induction, Divisibility and the Linear Diophantine Equation, much of which I have already encountered. The material is well written and the proofs are straight-forward but the exercises are a jump up in difficulty. I found myself struggling with a few and one I just couldn't see how to do at all. Still, it is to be expected in going up from level 2 to level 3.

I then started tackling the first book of M336 on Tilings and I must say that I think I like what I have read so far. It is refreshing to enter a new area of maths that I know nothing much about. This course also seems very well written and the ideas are being built up in nice easy stages.

In the mean time I have started on the second book of M381 which is on prime numbers. As this is something that interests me greatly, I will be relishing this book and am looking forward to its contents.

I have now signed up for M381 and M336 and my next aim will be to get together the funding I need to pay for the courses.