[r-t] Principles

Alexander Holroyd holroyd at math.ubc.ca
Thu Jul 7 02:15:51 UTC 2011

On Wed, 6 Jul 2011, Alexander Holroyd wrote:

> On Wed, 6 Jul 2011, Mark Davies wrote:
>> Yes, in particular, whilst it seems self-evident that all such examples 
>> must consist of sets of n bells, where each set follows the same path, and 
>> the n bells within each set follow the same path offset from each other,
> By definition this is true for n=1 and n=stage, but I'm guessing that's not 
> what you mean.  I don't at present see why it must be true for any other n.

OK, I think I can prove this, at least.

Suppose we have a method that is not a principle, but has all bells 
ringing the same blue line.  I'll call such a method an unprinciple. 
Define the lead to be the shortest possible block of repeating place 
notation.  Assume that the lead is not the whole course (this is a 
requirement according to the CC - can it be proved in this case anyway?).

Consider the lead end - ie the row at the end of the first lead.  Like any 
permutation, it can be broken into cycles.  E.g. the cycles of 53421 are 
(15),(243).  I claim that all the cycles have the same size (and by our 
assumptions, the size is not 1 or the stage).  If not, consider the row 
after ringing k leads, where k is the size of the smallest cycle.  The k 
bells in this cycle will have "come round", which means that the blue line 
repeats after r=k.(lead length) changes, ie any bell occupies the same 
place at any two times r apart.  If some other bells have not come round, 
this is a contradiction.

Now choose 1 bell from each cycle.  These form the first "set", as Mark 
describes.  Now see where they end up after one lead, to get the second 
set, and so on.

In particular this proves that unprinciples can only exist on composite 

> are there 
>> any examples where the the bells within a set are not in some sense 
>> contiguous, i.e. the structure is not reducible to a principle on M/n 
>> bells?
> I don't understand this - what is the reduction you are talking about?

I still don't understand what the reduction is.  Your description of 
"dewinking" does not fit either of the examples (e.g. in mine, 78 only 
ring together near the lead end).  However, I suspect that no construction 
like this (involving somehow "contiguous" bells) can account for all 
possible unprinciples...

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