[r-t] Principles

[Matthew Frye]

On 6 Jul 2011, at 09:54, Tony Smith wrote:
> The definition of principles that requires the number of leads to be the
> same as the number of bells, as well as all the working bells doing the same
> work in the plain course, dates from 1999 when the Decisions were amended to
> recognise Differentials (www.methods.org.uk/archive/ccm1999.htm). Frank
> Blagrove contrived the lead x1678x1678x18x18 which has all the working bells
> doing the same work but only 4 leads in the plain course.

Thanks you for the historical perspective and the example, so this is not the first time these methods have been considered.

On a more theoretical point, x1678x1678x18x18 has palindromic symmetry (more obvious when written 1678x18x-x), again worth noting that the symmetry points for line are offset from the symmetry point for the grid. I have a few further thoughts about the symmetry which I might post later.

I think it's worth noting at this point the complete list of symmetries as noted by Martin Bright (http://www.boojum.org.uk/ringing/symmetry.pdf):
• no symmetry; [demonstrated by Ander's nearly winked 16 bell version]
• vertical symmetry about a change, as Plain Bob; [Frank Blagrove's x1678x1678x18x18]
• rotational symmetry about a change, as Anglia Cyclic Bob Major; [Ander's original]
• vertical symmetry about a row †;
• rotational symmetry about a row;
• horizontal symmetry;
• horizontal and vertical symmetry about a change, as Mirror Bob;
• horizontal and vertical symmetry about a row †;
• ‘glide reflection’ symmetry, as Double Eastern Bob;
• vertical symmetry about a change, and rotational symmetry about a point
midway between the symmetry lines, as Bristol Surprise;
• vertical symmetry about a row, and rotational symmetry about a point
midway between the symmetry lines †.

Those marked with a † can't apply to a true method. They can, however, apply to the path of a single bell. If we can separate, to some degree, the symmetry of the line and the symmetry of the grid...

MF