[r-t] Falseness in cyclic methods [Was: Carlisle, Minor Music, Wraps, Cyclic focus]

[Richard Smith]

freepabs at cantab.net wrote:

> How do you define falseness groups in cyclic methods? (I'm sure you can)

There are two separate issues here: how do you define
falseness groups for glide-symmetric methods, and how do you
define falseness groups for irregular lead heads.

The set of false course heads for any method can be defined
as

  F = { a b^-1 : a, b in M }

where M is the set of rows in a course of the method.  (Note
I am using the convention that xy means x transposed by y
rather than the x transposition operating on y.)

Falseness groups are simply a way of partitioning F so that
if one element, f, of the falseness "group" is present in F,
so are all of the others.  Let's use the notation G(f) to
mean the falseness group containing f.

>From the definition of F alone, we know that if f is in F,
then so must f^-1.  (Proof: relabel a and b, and take the
inverse, (b a^-1)^-1 = a b^-1.)  Without knowing more about
the methods under consideration (i.e. do they all have
seven leads, are they palindromic, etc.), this is all we can
say about the falseness group structure.

  G(f) = { f, f^-1 }

Suppose that we now assume that all our methods are
constructed from leads with some lead head group, C.  (For
example, C might be the group of regular lead heads, or
perhaps the group of cyclic leads heads.)  We now know that
if the row r appears in M, then so must each of the rows
Cr.  This in turn means that if the false course head, f, is
present, so must each of CfC (recalling that C = C^-1).
This gives us

  G(f) = C {f, f^-1} C

And what about a palindromic method?  Well, if e is the lead
end change, then, if r is in M, er must also be in M.  This
means that we simply use the previous formula but replacing
C with the set {1, e}.  Or, if we are interested in
palindromic methods in multiple leads, then we can replace C
with the appropriate dihedral group.  For "conventional"
falseness groups, this will be the group of Plain Bob lead
ends and heads.

For all palindromic 7-lead major methods, the groups H will
be conjugate, and thus the falseness group structure is
essentially the same -- there will still be 28 "groups", and
they will still have the same essential properties.  So how
do we go about labeling the 28 irregular (e.g. cyclic)
falseness groups?  Put more mathematically, how exactly do
we define an isomorphism between the regular falseness
groups and the irregular ones?

It turns out that there is not a unique way of doing this --
there are, in fact, three distinct ways of mapping regular
falseness groups to irregular ones.  One way of looking at
this is that there is an ambiguity in defining the (plain
course) coursing order.  But more of that in a moment.



A falseness group can also be considered to be a
"permutation" of the coursing order.  Now coursing orders
are different from rows in that they are only defined upto
rotation -- that is so say, 7532468 is the same coursing
order as 2468753.  Actually we want to go slightly further
and say that 7532468 and 8642357 are the same coursing order
as they refer to the same course, albeit rung in different
directions.

To visualise this, we can imagine the coursing order written
around the vertices of a regular heptagon.  Two coursing
orders are the same if, by rotating and reflecting the
heptagons, they numbers can be made to match up.

Suppose we want to "permute" one of the coursing orders, say
by swapping a pair of adjacent bells.  We can't be any more
exact than this -- as the coursing order has no beginning or
end, we can't say we want to swap the second and third bells
in the coursing order.  All we can say is that they are
adjacent.  This means that the coursing orders 7523468,
7532648 and 7642358 are all generated from the plain course
coursing order by the same "permutation".  (The third is
clearer if you reverse the order.)

If we were to write down all of the coursing orders that can
be produced by this "permutation", we would find 7 different
ones.  Converting these into pairs of course heads and ends
gives the 14 false course heads of 'B' falseness.  (Try it
and see!)  Doing the same with other "permutations", produce
the other 27 falseness groups.  In this sense the 28
falseness groups are really just the 28 distinct
"permutations" of the coursing order.



During this discussion, I simply assumed that the plain
course coursing order is 7532468.  We all know this to be
true -- it's "obvious".  But *why* exactly is it?  We've
probably all heard the definition "it's the order the bells
go to the back", but that doesn't hold water -- just think
of fifths place bell Superlative.  A better definition is
some power of the place bell order.  For instance, taking
every 4th place bell from Cambridge Major gives the coursing
order.

The problem now is how do you decide which power of the
place bell order is the coursing order.  We need some
additional rule to tell us.  For instance, we could define
7-8 to be a coursing pair, or we could (usualy) require that
the bells affected by the bob are adjacent a coursing
triplet.  Sometimes, just looking at the method will give an
obvious answer.  However, it is quite possible for these
different definitions to give different answers.  And, in
general, there are three equally good answers.  (Three not
six because we've defined a coursing order to be the same as
itself backwards.)

It is this ambiguity in coursing order that makes labelling
irregular falseness groups ambiguous.  But, it also defines
an automorphism of the regular falseness group structure.
(An automorphism just means some transformation that
preserves the structure.)

  1 2 3           1 2 3           1 2 3           1 2 3
  -----           -----           -----           -----
  A A A           H U M           P C Y           c F L
  B a D           I I I           R d b           d b R
  C Y P           K E G           S X f           e O N
  D B a           L c F           T T T           f S X
  E G K           M H U           U M H           X f S
  F L c           N e O           a D B           Y P C
  G K E           O N e           b R d           Z Z Z

By relabelling from column 1 to column 2 (or 3) we haven't
fundamentally changed the structure of the falseness groups.
For instance, 'B' swaps adjacent bells in the coursing
order, 'a' swaps bells two apart, and 'D' swaps bells bells
one part, but what this actually means is dependent on the
choice of coursing order.



Well, so much for palindromic methods, but we know that
cyclic methods cannot be palindromic.  (Unless they're on
five or fewer bells or are offset cyclic, but we can ignore
that.)  What about glide-symmetric methods?

First of all we need a way of getting the reverse of a
change -- that is, given the 8-bell change 14, how do we
express the reverse change, 58?  Martin answers this in his
paper "Symmetry in Methods".  Writing reverse rounds as t
(tau in Martin's paper), the reverse of a change c is tct.

The sequence of changes in a glide-symmetric method will
then be of the form,

  (a_1, a_2, ..., a_n,  t a_1 t,  t a_2 t,  ...  t a_n t ).

The half-lead (backstroke at an even stage) row, h, can be
written h = a_1 a_2 ... a_n.  Using the fact that t^2 = 1,
the lead head can be expressed htht = (ht)^2.  Assuming
that, on n bells, the lead head is an n-1 cycle, this
implies that ht must also be an (n-1)-cycle.  (S_n is not
big enough to have other elements that square to an
(n-1)-cycle.)  More specifically, ht must be a member of the
lead head group, C.  Or, in terms of cyclic methods, all
glide-symmetric cyclic methods must have a reverse-cyclic
backstroke half-lead row.



What does this imply about the falseness group structure?
If r is a row in M (the method) then htrt must also be a row
in M.  (Proof: r = a_1 a_2 ... a_k, and t a_1 t t a_2 t ...
t a_k t = t a_1 a_2 ... a_k t = trt.)  The set of all
half-leads in the method will be Ct, so this means that if r
is in M so are Cr and Crt.

Unfortunately this is not very helpful when it comes to
fixed-treble falseness.  This is because each fixed-treble
false course head is produced when the treble is in some
specific place -- mathematically, f = a b^-1, where a and b
must have the treble in the same position.  On an even
number of bells, the treble must be in a different position
in Cr and Crt so we cannot use this to find additional FCHs
that must occur together.  From this point of view, it seems
that a general glide-symmetric method is no better than an
asymmetric one.



So what of assymetric falseness?  In general each falseness
group will split into four parts -- two in-course, two
out-of-course.  One might expect this to produce 4x28 = 112
different falseness groups; in fact only 71 are produced.
(This is for exactly the same reason that, while each
symmetric falseness group can in principle contain 56
elements, most do not.)  I'm not aware that there is any
convention for naming these falseness groups (though
presumably some numbering system similar to that used for
the splitting that occurs between Major or Royal would be
appropriate).

Phil-- I would be interested to know how you generated
falseness groups for these methods, and what naming
convention you were using.



Rotationally-symmetric methods are rather more complicated.
Several years ago, I generated a table of the falseness
groups for rotationally-symmetric cyclic methods.  (I can
probably dig out a copy if anyone wants it.)  Unfortunately
I can't locate a description of exactly how I did it.
(Phil-- did I sent you an email describing this, and you
still have a copy?  It would probably have been in the
autumn of 2002.)  As I recall, I found that each falseness
group split into an in-course and an out-of-course part, but
aside from that, there was no additional splitting.

One thing I do remember, however, is that it was not
entirely obvious which choice of coursing order to use to
definte the mapping between regular and cyclic falseness
groups.  In the end I decided that I wanted a 14 bob in a
2nds place method to produce a 'U' false course head.  This
meant defining the coursing order as 2345678 (or 8765432).

Richard