[r-t] Symmetry, bluelines and differentials

Alexander Holroyd holroyd at math.ubc.ca
Thu Feb 22 03:04:18 UTC 2007

> 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?

Unless I misunderstood something it seems clear that symmetric methods 
have exactly this property.  Symmetry means that if you reverse the 
sequence of place notations you get the same method (perhaps started in a 
different place).  Hence each bell's blue line is the reverse (possibly 
started in a different place) of some bell's blue line, perhaps the same 
bell or perhaps a different one, and there is a one-to-one pairing between 
original bells and reversed bells.

> 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?

The most notable property that breaks down is truth.  A symmetry point at 
a row means two consecutive place notations are the same, so with the 
exception of two-changes-in-the-plain-course aberrations like Cross 
Differential, the plain course is false.

See Martin Bright's article on method symmetry at 
for lots more info...

cheers, Ander

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