[r-t] 147 TDMM

[Richard Smith]

Simon Humphrey wrote:

> There are examples of non-course 5-lead splices amongst 
> the "irregular" non-Plain-Bob LE methods.

And indeed they do show up in some of the more complex 
splices within the 147 -- but never by themselves.  For 
example,

   720 Spliced Surprise Minor (3m)

     123456 Yo     - 134625 Yo     - 153462 Cm
   - 123564 Ip       142356 Ip       162345 Cm
   - 145236 Cm     - 163425 Ip     - 124536 Ip
     136524 Cm     - 154632 Ip       152643 Ip
     124653 Cm       165243 Yo       165324 Cm
   - 145362 Ip     - 165432 Ip     - 152436 Cm
     134256 Ip       146253 Cm       136245 Ip
     123645 Cm       153624 Ip     - 152364 Yo
   - 134562 Cm     - 146532 Yo       126543 Ip
     162453 Ip     - 146325 Ip     - 135264 Ip
     ---------       ---------       ---------
   - 134625        - 153462          123456

This is basically an irregular 5-lead splice between York 
and King Edward, with the K.E. entirely removed by inserting 
Ipswich and Cambridge in its place.  This is one of the 
compositions found in this search that is very definitely 
interesting, and I intend to write more about it when I get 
around to it.

> I see from your subsequent email you're considering 
> extending the search to include irregular methods. I 
> wonder if there are any other splices that don't exist in 
> the regular method domain.

We can easily list the types of splices that can exist 
between two methods (regular or otherwise) because we know 
that the lead heads and lead ends of the minimal unit of the 
splice (e.g. three leads for a three-lead splice) must form 
a group.

We also know that the group must contain an element with two 
sets of swapping pairs so that the splice works in whole 
leads.

Looking at BDP's 'The Composition of Peals in Parts', we can 
rapidly enumerate the possible types of splice by looking at 
all proper subgroups of A_5 ([5.02] in his listing).

   [4.02]  (A_4; order 12)
     gives rise to the six-lead splice

   [5.04]  (D_5; order 10)
     gives rise to the five-lead splices, including course
     splices

   [5.07]  (S_3, with two bells swapping for parity; order 6)
     gives rise to the three-lead splice

   [4.04]  ('pair of pairs' group; order 4)
     gives rise to two-lead splices

   [4.07]  (order 2)
     gives rise to the lead splice

The only unfamiliar ones are the two-lead splice and the 
non-course five-lead splices.

>> Unfortunately it is not possible to include both J and M 
>> variants in an extent (without also including other 
>> lead-end orders).
>
> This is true only if the search is limited to round block 
> extents.  A pair of J and M methods can easily be arranged 
> in a true and complete 720 that doesn't come round. Are 
> there any other cases where non-round block 720s might be 
> useful?

Quite possibly, yes.  Of the 4614 composition plans, 197 
cannot be joined up at all.  An example is the grid splice 
between Norwich, Westminster and Netherseale / Annable's 
London.

If that could be joined up in a non-round block, that would 
be useful.  It might, for example, be possible to produce a 
Bankes James type 1440 which had a true non-round-block
720 of Ws, No, Ab and Ne, followed by a true non-round-block 
720 of (say) Wk, giving a round-block 1440.

> Would extending the search to include these be feasible?

Certainly.  It would have a minimal effect on the run time 
of the search.  The code already finds all of these, the 
only change needed is getting it to print them out.

RAS