[r-t] Principles

Mark Davies mark at snowtiger.net
Wed Jul 6 21:02:04 UTC 2011

Ander writes,

> By definition this is true for n=1 and n=stage, but I'm guessing that's
> not what you mean.  I don't at present see why it must be true for any
> other n.

Nor I, but all the examples seen so far appear to be so.

> I don't understand this - what is the reduction you are talking about?

I'm just trying to get this straight in my head. How about putting it 
this way: there is a "de-winking" transform for these methods; a pair of 
bells (for n=2; triplet for n=3, etc) acts in a way analogous to a 
single bell at stage M/n.

Consider the behaviour of the front four of an 8-bell method of your 
type. Only certain sets of place notation are allowed. This is OK:

  2143 x
  2143 1234
  1234 x

Because it keeps the (12) and the (34) pair quite separate. The two 
pairs can therefore be considered to be two single bells ringing one 
change, in which both make a place:

  ab 12

Where a is equivalent to the 12 pair and b the 34.

It is also possible to make (12) and (34) hunt through each other, like 

  1324 14
  3142 x
  3412 14

This again reduces to the single pair a and b, this time crossing with 
each other:

  ba x

This transform (if it is such) applies to both Ander's and Frank 
Blagrove's methods. They both reduce, in this sense, to a Minimus 
principle with half the lead length but the same number of leads.

I have a feeling that any method which exhibits this property, which I 
shall call "slide self-similarity", or "slideyness" for short, can be 
constructed from a true principle at a lower stage, by winking up, or 
whatever the reverse of my reductionist transform is called.

But I don't know that for sure. My question is, are there slidey methods 
which cannot be constructed in this way?


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