[r-t] A tentative question about principles
Matthew Frye
matthew at frye.org.uk
Mon Jul 4 12:02:17 UTC 2011
On 4 Jul 2011, at 04:26, Don Morrison wrote:
> is it possible to have a method
> with no hunt bell(s) where all the bells do the same work, but there
> are not the same number of leads as bells?
On 4 Jul 2011, at 10:25, Simon Gay wrote:
> What about taking two leads of Original to be a single lead...
I guess this will hinge on exactly what you mean by "all bells do the same work".
If by that you mean "all bells ring the same set of place bells" then it's quite easy to prove (for any given place bell, 1 bell from the set "all bells" must ring it in lead one (by the definition of a place bell), so every place bell must be included in the set of place bells to be rung, 1 place bell per lead so no of leads = no of place bells = no of bells, this should be intuitively quite obvious).
If you mean "the blue line for each bell looks the same" it's a more subtle question. You start having to define what symmetry properties "looks the same" permits. If no symmetries are allowed, I believe it becomes the first case and is impossible. If you allow mirroring the line (either vertically or horizontally), then you can easily produce something (eg 3456x3456.18.1278x1278.18). With horizontal translation is also possible (eg ring major and put the same 4 bell principle on both front and back 4), with vertical translation, you can do what Simon suggests. I believe there is usually also a combined translation-reflection operation you can do (ie glide symmetry), which I suspect will also permit a solution.
Perhaps the more interesting question is if you allow vertical translation (of the line), can the method (grid) always be trivially expressed as a shorter block, repeated (as per Simon's example). I suspect so, but can't prove it.
MF
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