[r-t] ringing-theory Digest, Vol 52, Issue 7

[tom willis]

Given the set of all possible compositions of a given length in a given method, then define the 'distance' between two compositions as being the smallest number of block replacements to get from one to the other (oh look, a metric space). Then you're saying that any two compositions within a threshold distance of one another are the same - all well and good. 
However, if the distance between compositions A and B is within the threshhold, and the distance between B and C is within the threshold, that doesn't mean the distance between A and C is within the threshold. So while, A and B are the same composition, and B and C are the same composition, A and C are *not* the same composition. (In mathspeak "Distance not more than X" is not an equvilance relation due to lack of transitivity, providing X isn't zero). That's the key problem in my view.
 
You might argue that A,B,C are all the same composition because A=B and B=C... but one can keep adding new compositions to the pile, each of them suifficiently close to an exisisting one in the pile, until one ends with all the compositions in the set. So, either your threshold is zero, in which case every composition is different, or your threshold isn't zero, in which case every composition is the same.
 
Actually, something that's being ducked here is how big a block would we allow? If it's sufficiently small, then my all-compositions-are-one argument might not apply in some cases.
 
(This line of thought brought to you courtesy of long walk to the bank to deposit a cheque).
 
Tom Willis
 
> Message: 6> Date: Wed, 14 Jan 2009 20:03:07 -0000> From: "Mark Davies" <mark at snowtiger.net>> Subject: Re: [r-t] Ben Constant's Yorkshire Royal> To: <ringing-theory at bellringers.net>> Message-ID: <08e201c97683$2cc6a8e0$0400a8c0 at markx2>> Content-Type: text/plain; charset="iso-8859-1"> > Tom Willis writes,> > > Taking this to its logical extreme (as I tend to do), then surely any two> > peals of triples in a given method are the 'same', because you've just> > replaced one block of 5040 with another...> > Yes, and not just Triples. Many peals are simply plain courses extended with> blocks of three or whatever. But whilst we can take this to its logical> extreme, we don't follow it there. At some point, we say the number of> operations of additions or removals of blocks to get from one composition to> another exceeds a certain threshold, and we'll treat it as a completely> separate arrangement. In practice this is generally surprisingly easy to > judge.> 
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