[r-t] Principles

Alexander Holroyd holroyd at math.ubc.ca
Wed Jul 6 06:48:59 UTC 2011

Here at least is an answer to Matthew's last question: it is possible for 
one of these exotic beasts (non-principles with all bells doing same blue 
line) with no symmetry (on 16, not likely the minimum):



It is just the previous example winked up, with a couple of extra places 
added in the "2nds" (of the parent method).

On Tue, 5 Jul 2011, Matthew Frye wrote:

> 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
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