[r-t] Doubles Principles

[Matthew Frye]

On 16 Feb 2011, at 01:18, Samuel M. Austin wrote:
> Is it possible to have a Doubles Principle consisting of 10 or 20 changes per section?

The answer to this question is obviously yes, the question I presume you meant to ask was: can such principles generate extents? Again the answer is clearly yes (see Andrew's email), but I don't think this particular extent (or related ones you might easily find) is well described as a "principle". They are much better described as differentials.

This is because a 10/20 change division requires a 12/6 leads to generate the extent, the subgroups of  order 6 of the extent on 5 involve rotations on 3 and a pair swapping (3 different ways of putting these together to give a group of 6), subgroups of order 12 are either the alternating group on 4 or rotations on 3 plus a pair swapping (all combinations needed).
If you link the 10 change block using the alternating group on 4, you keep 1 bell fixed => plain method.
If you use and of the 3 rotating/2 swapping groups, I feel this is much more accurately described as a differential. Indeed there are some 36 differentials with 20 changes per division, all with lead ends 21453 & similar.

You may notice that I have assumed that the choice of leads must be generated by a subgroup of order 6 or 12. While this is usually what happens, I don't seen to be able to prove this necessary, even when the lead itself forms a subgroup, can anyone prove or disprove this? 
That assumption looks to be on even more dodgy ground when the lead itself does not form a group (I would think from memory that Ocean Finance Double ignores it? Need to check this one later).

So, in conclusion:
No, no one has rung one yet,
Yes, it's certainly "legal" to do so under the rules but that may be better described as a differential (IMHO),
Yes, it may be possible to construct something irregular that's well described as a principle,
No, I have no idea how you might go about doing that.

Sorry, I'm sure you were hoping for a shorter, simpler answer to your question,
Matthew Frye
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://bellringers.net/pipermail/ringing-theory/attachments/20110216/b580b165/attachment-0004.html>