[r-t] Hudson stuff
Mark Davies
mark at snowtiger.net
Sun Sep 19 21:55:20 UTC 2004
> a simple guide ... so everyone outside Cambridge, or not a
> mathematician, could at least get something out of this ...
> many of our best compositions came from non-mathematicians
> or Oxbridge graduates.
Clarrie, can I just point out that I read pure maths at Oxford, but I'm not
following any of this thread either. I did spend quite a lot of time in the
pub though.
I suppose I could venture a helpful definition or two. I expect I will be
wildly inaccurate.
When the mathematical geniuses of the list write "S_6" or "S_7" they mean
"the extent of changes on 6" or "7", etc. The "S" is because the
mathematician's group that corresponds to the ringer's extent is called the
"Symmetric group". I can't remember why, probably something obvious like
it's symmetrical, or they couldn't think of another name.
I guess everyone knows that ringing and group theory are closely related.
The reason is obvious if you understand what a group is. Basically, it's a
set of things, with an operation that operates on them. The set could be all
the whole numbers and the operation, addition of them (that's an infinite
group because the set is infinitely large). Or, it could be the set of all
rows on N bells and the operation of permutation (that's a finite group,
because even though the extent on 16 bells is quite large, it's still a
finite number).
Not all sets and operations make groups - integer numbers and multiplication
don't, for instance. That's because the operation has to follow three rules:
associativity, identity, and inverse. These rules make sure you get some
sort of structure where everything works out nicely. (I'm amazed at how
rigorously I've managed to put that.)
Associativity just means we don't care what order operations are done in,
i.e. a*b*c always gives the same result regardless of where you put the
brackets. (Numbers and multiplication is fine here). Identity means you can
find a "zero" element, which doesn't affect any other member of the set. For
instance, the number 1 under multiplication: x*1=x; for six bells, it's
place notation 123456. Finally, inverse means that for any element in your
set, you can always find something that reverses it, to give the identity.
Here's where numbers and multiplication break down: give me 0, and I can't
find anything to multiply it by to get 1. Ringing works though - given any
permutation of bells, I can find another permutation that'll get me back to
rounds.
It's amazing that something so simple can become so complex. But it sort of
starts like this...
Groups have subgroups. A subgroup has the same operation but only a subset
of the elements. For instance, all the changes with the treble at lead makes
a subgroup; also any "S_n" group has a subgroup half the size, which is all
the positive changes (or all the negative ones). You can make ("generate") a
subgroup by choosing a small number of elements out of the big group, then
multiply them every possible way until you get closure. Try it with 3412 and
2143 on four bells:
1234 (must have that, it's the identity)
3412
2143 (the two generating elements, call them a & b)
4321 (a*b, call it c)
There's no more elements you can get by multiplying a, b and c, so that's
your group - of order (size) 4. Try it with 3412 and 2134 - you should get a
bigger group of order 8. Some generating elements will give you the whole
group though.
Often subgroups are familiar as part ends of compositions. There's all sorts
of terminology built around them too: just one example - "coset". A coset is
kind of a partition of the big group, generated by the subgroup. If you look
at my little example above, 1234/3412/2143/4321, this picks out four changes
from S_4. But it also splits up the other 20 changes into cosets of 4
changes each. You make a coset by picking a change not in the subgroup (e.g,
2314) and multiplying it by each of the changes in the subgroup:
1234 x 2314 = 2314
3412 x 2314 = 4132
2143 x 2314 = 1423
4321 x 2314 = 3241
So the coset is 2314/4132/1423/3241. This isn't a group - it doesn't have
the identity for a start - but you can see it has a similar structure to the
subgroup.
Blimey, thinking about all that math has made me very tired. However I'm
looking forward to the inevitable array of corrections.
MBD
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