[r-t] Re: Irregular Falseness

[Richard Smith]

Robin Woolley wrote:

> This topic was aired with respect to Cardington S8 in the summer of '79 in
> the RW, (naturally). I haven't the details to hand but Roger Baldwin (I
> think) said, that for general coursing orders, he used the order in which
> the bells arrived in 7-8 at the lead end. Using this, Cardington has c/o
> 7523468

That's quite a good convention -- it's equivalent to
requiring 78 to be adjcent in the coursing order.

> It is obvious that falseness in irregular methods must be isomorphic to
> regular methods. Why? Consider the two methods K626 and K522 S8.

Someone having a bit of a musical joke, perhaps?

  K626   -36-6-58-36-14-38-36-18 lh 12    13527486
  K522   -36-6-58-36-14-38-36-78 lh 12    15478263

So K626 has the regular coursing order, 7532468, while K522
has the irregular coursing order 7542368 (using Roger
Baldwin's convention).  The first section (when the treble
is in 1-2) generates the false lead head 2463578 (which
happens, conveniently, to be a course head).  This
corresponds to (regular) falseness group, E.

This FCH corresponds to a 4-cycle (3,5,6,4).  Imagine the
both coursing order written around the vertices of regular
heptagons, and imagine this 4-cycle drawn on the heptagons.
In the regular case, the 4-cycle is a trapezium; in the
irreguar case, a pair of triangles:

  5              6          5               6
   -------------              -------------
   \           /               \         /
    \         /                  \     /
     \       /                     \ /
      \     /                      / \
       \   /                     /     \
       -----                     -------
     3       4                 4         3

Even allowing for the transformations described in my
previous email, you cannot deform these two shapes to be the
same.  (This is not as obvious as it might seem;
nevertheless it is true!)  This means the falseness groups
are necessarily different.  So what is the irregular
falseness group?  From the geometry, it is the same as the
regular falseness group containing the 4-cycle (4,5,6,3),
which is the FCH 2634578, a member of the F falseness group.

So the regular method starts with E falseness, but the
irregular one starts with F.  As F has 14 members and E, 28,
this clearly cannot be due to some isomorphism between E and
F.

> These have
> common place notations apart from the half-lead place so K626 is regular and
> K522 not. Since falseness can be derived from just the first half-lead of a
> method, then these two methods must have the same falseness - it just looks
> different - but that is what isomorphism deals with.

No.  There is, in general, no isomorphism between the
falseness groups of a regular method and of an irregular
half-lead variant.

In principle, it's quite possible to have a regular method
where each section produces the same falseness group, but
for its irregular half-lead variant to product different
falseness groups from each section.  (Friday challenge: can
anyone find an example of this?)

> Also, from memory, during the discussion of notation of lead end groups for
> irreg. minor methods, weren't those which have 4ths place lead ends missed
> out?

Yes.  I'm not aware of any standard notation for these.
Michael Foulds uses X and Y for two of them in his books on
spliced TD Minor.  I guess A-F are free for the remaining
six.

> (Nice to see the 'traditional' letters 'G' - 'O' used).

For minor, I prefer these to the more modern a, b, e, f, g,
h, l and m.

Richard