I am progressing ok. I finished unit 3 of the number theory half of M381 on congruence. This was fairly straight forward and didn't offer any particular difficulties. In the mean time I have also finished unit IB3 of M336 on frieze groups and again this wasn't particularly challenging. I can now see where this course is going with its classification of friezes and wallpaper patterns. I did always wonder what Alan, our tutor for M208, was always going on about when he said that there were 17 different types of wallpaper pattern and now I have an inkling of what he may mean. I have also now got through a fair chunk of unit 1 of the mathematical logic half of M381 on computability. This is ok as it is basic linear programming.
Going back to friezes, I thought I would show you an example of a frieze that can be found in our flat.
My Dad (who knows these things because he is an architect) pointed out to me that this type of cornice is called "egg and dart" which I think is an apt description. From a mathematical point of view, if you take the decoration bit in the middle as the frieze (ignoring the fact that it obviously doesn't extend indefinitely to the left and right), then it has a number of different symmetries. For example, if you take a basic element of an egg bounded by half a dart on either side, then you can translate this element horizontally left and right by multiples of the distance between two adjacent darts. This is what defines a frieze. There are also, obviously, reflections in a vertical axis (either through a dart or through the centre of an egg). There are, however, no rotational symmetries or reflections or glide reflections in a horizontal axis.
Each frieze can be analysed according to its group of symmetries and an algorithm is given in the unit for identifying the type of frieze according to its group. This is a type 2 frieze or in international notation a pm11. There are seven types of frieze in all, as defined by their symmetry group.
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