[r-t] More from the method-masher - "impossible" triples principles

[Simon Humphrey]

It was Simon Melen.

The date on the spreadsheet I have which contains this extent is dated
22.7.2014 if that's any help.

 

SH

 

From: ringing-theory <ringing-theory-bounces at bellringers.org> On Behalf Of
Alexander E Holroyd
Sent: 22 July 2022 00:42
To: ringing-theory at bellringers.org
Subject: [r-t] More from the method-masher - "impossible" triples principles

 

A few years ago someone published a remarkable 1-part extent of pure triples
with no structure whatever - to all appearances a random string of place
notations 1,3,5 and 7.  Moreover this was derived by a fascinating search
algorithm involving starting from any extent and successively modifying it,
with a key element being reversing sections when falseness is encountered.
I cannot find this original posting, nor can I remember who did this.
Please can someone enlighten me, so that they can have the credit they
deserve for this very interesting idea?

 

Anyway, I have explored extensions of the idea, as I'll explain below.  I
propose to call the algorithm the method-masher.  My main discoveries so far
are:

 

1. Any combination of place notations can be used (so long as there are at
least 3).

2. It is possible to get a 7-part structure, i.e. a principle whose plain
course is the extent.

3. One can seed the process with *any* extent, or 7-part extent accordingly,
not necessarily composed of the desired place notations.  (I used
https://complib.org/composition/98367).

 

To illustrate this, here are example principles with place notations 1,3,7
(so pure triples with only 3 place notations), and with 1,345,7 (which have
a nice symmetry, and no long places possible), and with 3,147,5 (ditto).

 

https://complib.org/composition/98370

https://complib.org/composition/98372

https://complib.org/composition/98371

 

This might be the first known 7-part extent using only place notations
1,3,7, for example.  (Please correct me if not).

 

Further thoughts and possible limitations:

 

1. It might be possible to get a symmetric principle this way, although it
seems a bit more tricky.

2. There seems to be no way to avoid long places other than choosing place
notations that cannot give them, so the approach is of limited use for
Minor, for instance.

3. The approach seems only useful for generating methods, or compositions of
methods with no internal falseness.  So no use for producing new
compositions of Stedman, for example.

 

Here are the details.

 

The basic object that we work with is an "extent that does not come round",
i.e. a path consisting of all 5040 rows connected by 5039 changes, but with
no final change to take it back to the start.  In mathematical terminology
this is a Hamiltionian path.  The start and end do not need to be rounds.
If we are starting with a proper extent that does come round, we can get one
of these by simply breaking it apart across an arbitrary change, and
considering it to start there.

 

In 7-part world, we instead use a Hamiltonian path in the Schreier graph,
i.e. a sequence of 720 rows connected by 719 changes that give the extent
when pre-composed by the group of 7 part ends.

 

Given such a path, from row x to row y say, we choose a place notation to
add to the end of it, after the y.  This takes us to a row we already have,
say z.  We modify the path so that after the first visit to z we follow the
new place notation to y, and then follow the rest of the path backwards
until just before we hit z again.  This gives a new path, and we repeat the
process.

 

To avoid going nowhere, the new place notation chosen should be different
from the last place notation before y, and also different from the
placenotation that was erased at the end of the last step.  But with 3 place
notations to choose from this is always possible.

 

It is possible that z = x, in which case we have found a round block!  If we
have done enough mashing we can stop, or if not we can split it in a random
place and keep going.

 

We can start with any extent at all, or any 7-part extent, like
https://complib.org/composition/98367.  Eventually all the original place
notations get replaced with the ones we want.

 

With only 3 place notations, there is no choice at a typical step, because 2
of the 3 place notations are disallowed.  I found that occasionally the
process would get stuck in an endless loop and never produce a round block.
When that happened I simply backtracked and tried splitting the previous
round block in a different place.

 

cheers, Ander

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