[r-t] Symmetry, bluelines and differentials

Don Morrison dfm at ringing.org
Wed Feb 21 16:02:15 UTC 2007

For normal methods, possessing the usual symmetry about a half-lead in
the changes results in the blueline for all the working bells also
possessing a similar symmetry.

But in differentials this isn't quite the case. For example, consider
the differential formed by having cross instead of places at the lead
end and half lead of London Major:

  3x3.4x2x3.4x4.5.6x6x  lh x

The blueline rung by the cycle of working bells {1, 7, 6, 4} is not
symmetric. Neither is that rung by the cycle of working bells {2, 3,
5, 8}. But they are reverses of one another.

Presumably there is some relationship between the usual symmetry of
the changes and some kind of symmetry in the blueline(s). What is it?
Is it perhaps

All sub-bluelines are either symmetric, or appear in pairs that are
reverses of one another?

Or is it something more complicated than that?

The usual symmetry involves reflection about about two changes spaced
half a lead apart. There's not reason both such reflections couldn't
instead be about rows, or (if the lead length is odd) one about a row
and the other about a change. Are there any expected properties that
break down in either or both of these cases?

I, at least, tend to think of the usual symmetry as the changes are
the same in both directions, when reflected about the lead end and
half lead. This doesn't work if we extend change ringing to include
jump changes. Is a more appropriate definition of symmetry in that
case that we both reverse the order of the changes, and replace them
all with their inverses (in the absence of jump changes are changes
are self-inverse)?

Don Morrison <dfm at ringing.org>, <dfm2 at cmu.edu>
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