[r-t] Interesting observation

Mark Davies mark at snowtiger.net
Tue Oct 18 07:28:42 UTC 2005

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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