Well I was dreading this but I got one of my highest scores so far. I do think my tutor Alan has been quite generous to my astonishment I only dropped 1 mark on the last question. Still my rant against the emphasis on visualisation seems to have sparked off some interest in other people who have been inspired to come up with a non visual way of dealing with rotating flags etc.

The essence appears to be

1) Work out the effects of the symmetry operation of the group in terms of permutations of the areas

2) A colouring can be seen as a mapping from each square to the set of N colours {B,W...}

3) So and this has yet to be clearly defined the number of fixed objects is related to the cycle stucture of the permutation raised to the power of the colours. But you must include all the cycles.

4) One of the guys on the M208 forum Toby has suggested that the nmber of fixed points for a given symmetry operation is

$$ |C|^{nc}$$

where |C| is the number of cycles associatied with each permutation and nc is the number of colours others have been trying to prove or disprove this. My mate Neil thinks he's close to finding a general algorithm

Whatever the outcome, it's good that people are 'playing' with the Maths and not just desparately trying to find answers to TMA questions.

Good skills Chris. It's also nice to see that M208 offers enough variety and deep enough maths skills, to allow people to start hunting around for rules and algorithms to fit general cases, without being taught the proofs etc. Have you or anyone else yet, caught sight of the M303 2013 course syllabus, in any detail?

ReplyDeleteIt will be interesting to see if the condensation of 3 x 30pts into 1 x 60pts at level 3, has lost any of the depth of the subjects involved.

An excellent result. Well done, Chris!

ReplyDeleteI hope the Cambridge project is going well.

No I haven't seen the actual syllabus but from what I can gather

ReplyDeleteYou lose

1) Any hint of Godel's theorem or logic

2) Any hint of continuity as applied to topological spaces (crucial as far as I can tell if you want to understand the deep theorems that Penrose and Hawking developed in understanding the singularities associated with General relativity)

3) Any tie up between Group theory and geometry

In compensation you get a bit on ring theory and fields

In an ideal world a new 30 point course would have been introduced covering rings fields and modules and none of the core material would have been dropped.

Ok I 'll just about squeesze everything in but I think it's really unfair on those who haven't got their timing correct.

And of course I would have loved to have done the options available 20 years ago namely M203 which covered linear analysis and M202 which covered topics in pure maths but of course this is not avaiable now or Lebesgue Integration or differential geometry

It goes without sayin]n that to you and anyone else I'll send the relevant pdf's on request but it will never be the same as actually studying the course doing the TMA's

It really does seem as if the OU wants to offer less for more and I'm not sure I understand why

Ha ha

ReplyDeleteI dropped 5!

The general algorithm is far, far away and maybe mythic I'm afraid. It gets further away the more I think about it.

Toby is right about the number of cycles, but generating the permutations is difficult.

If we think about a triangle how many ways can you divide it symmetrically?

I'll see you next Saturday hopefully.

arb

neo