[r-t] Principles

[Matthew Frye]

On 5 Jul 2011, at 22:07, Alexander Holroyd wrote:
> -123458-1238-1678-145678
> http://www.boojum.org.uk/cgi-bin/line.pl?bells=8&pn=-2345-23-67-4567&lines=1&line=2&line1=b&line1c=red&line2=b&line2b=2&action.x=1
> 
> I find it fairly amazing that such a thing can exist.  Is there a general theory of such methods?  Can someone come up with a nice one that people would want to ring (preferably with a nice peal compositions available)? What is the smallest number of bells on which one exists?

Very impressive.
One thing I notice: this has translational symmetry, but it also has rotational symmetry of both the line (by itself) and the grid, the interesting thing is the placement of the symmetry points on the line, 1 row before and 1 row after the symmetry point of the grid (there are actually 4 points on 2 lines, but when overlaid on the grid, pairs of points from the same line lie on top of each other), the rotation about the grid symmetry point relating the two line symmetry points (and in general the 2 separate lines, obviously).
Is this inevitable from the combination of translational and rotational symmetry? Intuitively I think yes, as the two symmetry points on the lines have to be equivalent and so related through symmetry. Can we use this to construct other methods?

Further questions: is a similar construction possible in the presence of other symmetries? I think we've already ruled most out, but glide symmetry perhaps? What about no symmetry other than translational?

MF