[r-t] New mirror principles

[Philip Saddleton]

On Thu, 2021-02-04 at 21:56 +0000, Richard Pullin wrote:
> Methods with mirror or horizontal symmetry are still quite rare.
> These involve pairs of bells ringing mirror blue lines of each other
> in perfect unison. To achieve this you must use horizontally
> symmetric place notations, which in Minor are limited to 16, 34,
> 1256, X. The plain course of Kidderminster Minor sweeps up all 48 of
> the mirror rows on six bells - the mirror pairs being bells 16, 25,
> 34 - and to produce a 720 you simply need to visit all 15 of the
> different possible pairings.

I have only just come across this thread - interestingly I had been
having simlar ideas, though from the point of view of getting extents
of spliced.

> 
> I think principles make the most elegant mirror methods as treble-
> dominated methods require the treble and tenor to both be hunt bells,
> which seems a bit bizarre. 
> 

Arlesey Bob Minor <https://complib.org/method/12035> has exactly the
same rows in a plain course as Kidderminster. So in a 720, assuming the
calls are compatible you can replace a whole course of one method with
the other. The possibilities for methods of this type are fairly
restricted though. Are there enough sensible ones (e.g. not a one-lead
course) to get an extent in 15 methods?

> On eight bells there are 384 mirror rows, the permitted place
> notations being 18, 36, 1458, 1278, 3456, 123678, X. The nicest of
> these are 18, 36, X, but it seems you can only attain 192 rows using
> just these place notations, in much the same way that you can only
> ring 60 rows in Doubles without using single changes. 

The reason for this is that the changes that the even change (X) keeps
bells in the same half of the row, while the odd ones swap a pair in 4-
5. So the possible rows are the in  course ones with an even number of
bells in the half they started in, and the out of course ones with an
odd number.

I came up with
this: https://complib.org/method/41162?accessKey=a829c83566f003025dcb638e01a32df35e3064e1
 - which I believe is the only possibility for a palindromic method
with Original lead heads. It has similarities with Quick Six Triples:
ring quick sixes at front and back with double dodging in the middle,
then replace 18 with x when it would come round.

I struggled to think of an elegant way of selecting courses that could
be joined with 3 member Q-sets. Would a 5-part or 7-part be possible
with more than the mimumum linkages?

Anyone up for a 40,320 of spliced in 105/210 methods, atw and eld for
all 8 bells?

PABS
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