[r-t] Non Plaindromic Plain Minor Methods
Robin Woolley
robin at robinw.org.uk
Tue Jun 26 03:29:49 UTC 2012
Hi All,
Harold Chant is on record as saying that "It may well be as well to stress
that all asymmetrical Minor methods are, under ordinary circumstances,
false." (RW1965/12/17)
This is not true. In June last year, I conducted the first extent in an
asymmetric plain minor method with regular lead ends and heads using only 21
comon bobs and singles. Given that the standard extent requires twenty such
calls, it seems that this was a completely "ordinary circumstance".
Coevally, and unbeknownst to me, Michael Foulds was researching the complete
set of CC recognised 'non-palindromic' plain minor methods. He showed there
are 1412200 such methods. 649348 having conventional double change lead-ends
(12, 14 or 16). Using 'Smith's Theorem'
(www.methods.org.uk/other/asl7111.htm), these reduce eventually to 312
methods which should,in theory, be able to produce an extent with no more
than two blows in each place using lead-end calls only.
At the time of writing, using the 'required lead head' method described
previously on this list, extents for 174 methods have been found - these
falling into 30 groups. (These are real mathematical groups).
These groups are isomorphs of Z2, Z3, Z4, Z6 and S3. Not all of the
isomorphs have yet been found to produce extents such as the Z6 isomorph
generated by 35624 for group H methods.
However, when the groups isomorphic to group V are considered, no extents
have been found. For example, the group, which has been arbitrarily named
G006 = {24536, 53426, 54326} No extent has been found for this - using the
lead-end-only call criterion.
Is there a theoretical reason for there being no extents for group V
isomorphs - or is it just the unfortunate juxtaposition of required lead
heads and inevitable lead ends? The only distinguishing feature of the
V-isomorphs is that it is necessary and sufficient that they contain a
4-fold transposition between corresponding rows. This is as follows. A
method starting with place notation 'x' has second change 214365. A Group G
method ending 34 has penultimate change 312564. From 214365, 312564 is a
tranposition of order 4.
We have discovered quite a bit about the properties of these methods. For
instance, a group J method with RLH 32465 cannot have common bobs in an
extent. It would be useful to know the answer to the question above and I
would especially be interested in an extent for the group G006.
By the way, the method we rang was from group "K046" which has RLH 43256 and
the comp. was: W, sW, IH, 4OsW x 3.
To sum up, is it theoretically impossible to find an extent where the
Required Lead Head group is isomorphic to (Klien's) Group V?
Best wishes
Robin
More information about the ringing-theory
mailing list