[r-t] Hudson's Group and A_5

Fri May 15 15:16:37 BST 2020

> From: Richard Pullin
> To: ringing-theory at bellringers.org
> Date: 14/05/2020 21:37
> 
> Recently I produced some 60-part extents of twin hunt Triples 
> methods, using Hudson's group 'Hud60' as part ends. These can be found 
here: 
> https://complib.org/collection/10788 
> 
> 
> As has been discussed on this list and in a Brian Price paper, there
> is a dual correspondence between A_5 and Hud60, due to the unique 
> outer automorphism of S_6. The rotation symmetries of an Icosahedron
> can produce A_5 and Hud60, a description of how to do so being 
> posted on this list in April 2008. Are the two peals by Lockwood and
> me are a rare example of this correspondence being obvious in a 
> ringing context? Usually internal falseness gets in the way, like 
> the Stedman sixes working against A_5 part ends yet still being so 
> ideal for Hud60. But in the P-Blocks of twin hunt methods there is 
> no in-course falseness at all, so no favouritism towards either group.  
> 
> Other examples of A_5 meeting Hud60 in ringing composition. Hudson 
> Delight Minor (aka Norwich Delight) has been discussed on this list 
> - a half lead of the method traverses the cosets of Hud60, thereby 
> making variable treble 720s possible based on this group. But you 
> can still call W,H,W to make an A_5 extent, with the unaffected 
> treble being the ' fixed observation bell.' 
> 
Another example of the dual correspondence between A_5 and Hud60,
between Stedman Triples and London Surprise Minor was pointed out 
by Philip Saddleton in 2003.

I wrote on change-ringers 23 January 2003:
actual email missing at:
https://lists.ringingworld.co.uk/private/change-ringers/Week-of-Mon-20030120/083919.html

>From my notes [from 1994/1995]:

These two peals are interesting in that they are based on only two 
courses,
SH and HL.  No calls are made at Q, and there are only two singles, one at
14
and the other at 2. These are the only peals on this plan, apart from
trivial
variations of start point, reversals, or which courses have the singles.
I did not see any peals based on this precise idea in the Stedman Triples
collection.  Perhaps the closest is that of J Carter (Number 21 in CC
collection), but that has a lot more singles (22 in total).
The second halves of both compositions are the same as the first halves,
with
HL replacing SH and SH replacing HL.  The start of second half is offset
from
the first half.  It may be possible to chose the position of the singles 
to
give a second half which starts at the same point as the first.  Similarly
it
may be possible to find a starting point with SH instead of HL, while 
still
preserving the single at 14 half way and single at 2 at the end.
The first peal has quite a simple construction, reminiscent of a 360 of
Plain Bob Minor called WHW;WHW;WHW (from the 5th).  The second peal is 
more
complex, and appears to follow no set pattern.
A computer search reveals there are 160 half peals with courses of just SH
or
HL, and no longer touches.  These two peals illustrate all those possible
half
peals.  The first peal can be split into two halves.  Each half can be
started
in 10 places, and can be rung forwards or backwards.  This gives 40
possibilities.  The second peal can be split into two halves.  Each half
can be
started in 30 places, and can be rung forwards or backwards.  This gives
120
possibilities.  Altogether this gives the 160 half peals.

231456  2 S H L 14
__________________
641235      x x   |
521643      x x   |
361524      x x   |
451362      x x   |
562314    x x     |A
432561      x x   |
152436      x x   |
642153      x x   |
453126    x x     |
526134    x x     |
__________________
354621      A
164352      x x
234165      x x
514236      x x
624513      x x
213546    x x
653214      x x
423651      x x
163425      x x
625431    x x
(524361)   x x    s
__________________
134526      x x   |
654132      x x   |
532146    x x     |
346125    x x     |
425163    x x     |B
263154    x x     |
413265      x x   |
165234    x x     |
634251    x x     |
351246    x x     |
__________________
524361      2B
231456  s
__________________

231456  2 S H L 14
__________________
641235      x x
521643      x x
243615    x x
415632    x x
132654    x x
462135      x x
512463      x x
163425    x x
543162      x x
213546      x x
653214      x x
514236    x x
136245    x x
526134      x x
416523      x x
356412      x x
246351      x x
451362    x x
562314    x x
432561      x x
361524    x x
624513    x x
354621      x x
164352      x x
234165      x x
365142    x x
215364      x x
435216      x x
625431      x x
(524361)   x x    s
261345    x x
645312    x x
412356    x x
632415      x x
542631      x x
431625    x x
325614    x x
214653    x x
153642    x x
263154      x x
413265      x x
165234    x x
425163      x x
315426      x x
126453    x x
253461    x x
561432    x x
241563      x x
463512    x x
612534    x x
134526    x x
654132      x x
532146    x x
346125    x x
516342      x x
236514      x x
456231      x x
531264    x x
364215    x x
524361      x x
231456  s
__________________

[End of quote from 2003]

Actually both peals were originally composed by J W Parker.
https://complib.org/composition/45193
See Bell News volume XXV No. 1290 of December 22nd, 1906, p. 456. 
https://complib.org/composition/28888

Philip then replied on 26 January 2003:
https://lists.ringingworld.co.uk/private/change-ringers/Week-of-Mon-20030120/083944.html
I did an analysis of this some time ago, prompted by Andrew, but I can't
find my notes. The resemblance of the first to the standard calling of
minor, and the fact that the calling of the second half is the same as
the first with SH and HL interchanged are no accident:

To ensure that only SH and HL callings are used, the Q-sets have to be
grouped into sets of three, i.e. the courses from 231456, 356412,
563241, 461253, 416523, 213546 have to be called identically.

It is well-known that Hudson's 60 courses are isomorphic to the 60
in-course rows on 5 bells, and this demonstrates the isomorphism. Take a
touch of London Minor, and replace each plain lead with SH and each
bobbed lead with HL, and a touch of Stedman Triples will result. The
Q-set rules in the two cases correspond, hence the longest true touch
corresponds to 720 of Minor, i.e. 30 courses, or half a peal. This
contains precisely half of each of the sets of 6 courses: the other half
correspond to the same leads of Minor rung backwards. Thus there is a
second mutually true half peal with the courses rung in reverse order.
Joining the two halves with singles reverses one of them, putting the
courses in the reverse order and swapping HL and SH.

---
Andrew Johnson
Twyford

Unless stated otherwise above:
IBM United Kingdom Limited - Registered in England and Wales with number 
741598. 
Registered office: PO Box 41, North Harbour, Portsmouth, Hampshire PO6 3AU