[r-t] Interesting observation

Philip Saddleton pabs at cantab.net
Tue Oct 18 17:56:30 UTC 2005

I thought that this was the explanation that I would have given, until I 
read it more closely. The point that I would make is that the coursing 
order after the single has cyclic symmetry - there are six such, i.e. 
rounds, queens, tittums and their reverses

8765432 8234567
8357246 8642753
8526374 8473625

Ringing a lead of any regular method in any of these courses has the 
effect of a cyclic rotation of the working bells. So undoing the 
transposition that got you there from the plain course after such a 
rotation gets you to a course with a cyclic lead head.


Mark Davies said  on 18/10/2005 08:28:
> I wrote:
>>This is something to do with the symmetry between the 2nd and 8th as
>>the discontinuities of the cyclic leadhead with respect to PB.
> Aha - and the other vital point of information to form a complete
> understanding of this, is to notice that the cyclic coursing orders are
> formed from shifts and reversals of the natural coursing order; furthermore,
> the shifts can be accomplished by overlapping reversals.
> Because reverses are self-inverse, it proves possible, on any even number of
> bells, to use just two reversal operations to obtain any of the cyclic
> coursing order. The minimum size of the reversal operation needed is n/2-1,
> where n is the stage. In other words, the operation must be sufficient to
> reverse all of the odd bells in one go. On eight this means a 3-reverse:
> 8753246 -> 8357246
> Which looks like a common single, of course. On higher numbers, a larger
> minimum reversal operation is needed. So for 12, a 5-reverse:
> TE975324680 -> T3579E24680
> You can see how this one operation can generate all the other cyclic COs by
> bringing one bell through at a time. After reversing the odd bells, we can
> choose to bring either 2 or tenor through; here is 2, requiring 4,6,8,0,T to
> be reversed:
> TE975324680 -> T3579E24680 -> 3579E2T0864
> If you bring the 3 through next, one of the previous reversals is always
> backed out:
> 3579E2T0864 -> 32E975T0864 -> 32E9754680T
> Hence this transformation can be achieved from the natural CO in just two
> operations, one of which is always the reversal of the odd bells:
> TE975324680 -> T3579E24680 -> 32E9754680T
> Obvious, really.

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