[r-t] Ringable Alternating Groups On Five Bells
grandsirerich at googlemail.com
Sat Aug 29 00:26:37 BST 2020
This is really interesting and useful, thanks for sharing.
As the alternating group on 5 is isomorphic to Hudson's group, it would be
interesting to see which of these blocks can be turned into Hudson type
components for Minor methods and principles. For example, the Doubles six:
220.127.116.11.3.1 can be replaced by 18.104.22.168.34.1 in Minor, which is a very
easy-to-see demonstration of the (123) cycles being replaced by (123)(456)
cycles. I'm sure the idea of using A_5 in this way will have occurred to
others, but they may have been put off by the tendency of 3/4-blow places
in the resulting Minor methods.
If you arbitrarily try this on a bobbed lead of Grandsire Doubles, one
result is: 22.214.171.124.34.1,2 (which perhaps closer resembles Double Grandsire
Doubles with bobs at the half lead and lead end.) The method's plain course
is Hudson's group, so the trick has worked. As the half lead rows are the
same as in Cambridge S and Oxford TB, you can get a number of variants by
replacing the half lead with 5 and/or the lead head with 1. I was then
reminded that one of these variants had already been devised a few months
ago by the mathematician Robert A Wilson (though his method has the tenor
as hunt bell, so I rotated it.)
We can go a step further and turn this into a challenging Principle with
360 changes in the plain course, though some might prefer to think of it as
a variable-treble touch of the Plain method:
a 720 you simply use two singles a course apart:
I love how this extent is analogous to a 2-single 120 of Stedman Doubles,
which brings us right back to the original topic of A_5.
So what else can be done with the Doubles blocks?
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