[r-t] All runs in a quarter, proper version
Alexander Holroyd
holroyd at math.ubc.ca
Mon Feb 2 04:48:41 UTC 2015
With less conventional place notations (1234 etc) and more static music,
one can do the same thing in a 1280:
http://www.math.ubc.ca/~holroyd/runs2.pdf
Less good, in my opinion, although certainly ringable. Considering the
absolute theoretical minimum is 736, I find it pretty amazing that this is
possible.
On Sat, 31 Jan 2015, Alexander Holroyd wrote:
> I'm pretty excited about how well this 1536 turned out! I for one would be
> keen to ring it. Here is the whole thing written out, with the runs
> highlighted:
>
> http://www.math.ubc.ca/~holroyd/runs.pdf
>
> I've put together a description of the process that went in to composing it.
> In particular I hope this will answer Philip's questions, and perhaps
> encourage others to explore this interesting new paradigm.
>
> I started out with the idea of trying to get all 480 front and back 4-runs
> into a reasonably short length that someone might actually want to ring.
>
> How closely packed can the runs be? Of course one can repeat a run
> immediately with e.g. a 5678 place notation:
> 12345678
> 21435678
> but while that's a potentially interesting effect, I decided I wanted to try
> to stick to more conventional local structure (e.g. maybe all triple and
> quadruple changes). Of course one can repeat runs every other row using
> dodging:
> 12345678
> 21436587
> 24135678
> 42316587
> 43215678...
> but more than a triple dodge or so would get boring, and one must link to
> other types of run. In a few cases one can go directly to another type, e.g.
> 21435678
> 12346587
> or
> 24315678
> 23451768
> but there are not many such opportunities. Many more linkages open up if one
> allows things like the following, where two run-rows are separated by two
> other rows, with both the run type and direction changing:
> 12834567
> 21384657
> 23148675
> 32418765
>
> Based on this, it might be reasonable to have no more than two other rows
> between two run-rows in the entire touch. However, I couldn't think of an
> obvious way to impose enough structure to make this into something sensible
> rather than a long random string of place notation.
>
> An obvious solution to the structure issue is to aim for a cyclic 8-part. It
> is no longer reasonable to expect the property above, but instead one can
> still hope for at most two other rows between potential run-rows, i.e.
> run-rows plus things like -7812 that will become a run-row in other parts.
>
> My first attempts involved starting from simple basic blocks. E.g. here is a
> natural set of round blocks that together contain all the runs, if rung 8
> times each according the cyclic part ends 23456781 etc.:
> 12345678: -4.5.6.5.4-4-4-3.6.3-4-4 *3
> 13245678: -4.5.6.5.4-4-4-3.6.3-4-4 *3
> 12345678: -5.4.3.4.5-5-5-6.3.6-5-5 *3
> 12345768: -5.4.3.4.5-5-5-6.3.6-5-5 *3
> The problem is that these are false against each other in multiple places.
> E.g. rounds and 43215678 each occur in two blocks because they each contain
> two runs, but so does 43216587 because it appears both as a 1234 run and as a
> link row for 5678 runs. This kind of thing seems inevitable whenever one
> starts with simple repeating blocks. Falseness is not necessarily a problem,
> because one can eliminate it by local changes, and potentially even use it to
> link blocks together. However, the very nature of the shared runs issue
> requires different solutions in different places. E.g. starting from the
> above blocks, here is a true set of round blocks I found that contain all the
> runs in a cyclic 8-part structure, retaining rotational symmetry (more on
> that later):
> 43125678: 5.6.5.4-4-4-6.3-5-5-5.4.3.4.5-5-5-3.6-4-4-4
> 32145678: 4-4-4-3.6-5-5-5.36.45.36.4-4-4-3.6-5-5-5.36.45.36
> 42135678: 56.1-5-5-5.4.3.4.5-5-5-1.56.4-4-4-36.1.36-4-4-4
> 34215678: 1.34.5-5-5-36.1.36-5-5-5.34.1-4-4-4.5.6.5.4-4-4-
> 21438765: 4-4.5-5.4-4.5-5
> 43216587: 4-4.5-5.4-4.5-5
> 43215678: 4-4.5-5.4-4.5-5
> 43218765: 4-4.5-5.4-4.5-5
> These still need to be joined together, and the 8 parts need to be joined
> somehow. One could certainly do this, but it would probably end up not very
> elegant, with little sign of the original structure. Getting an exact 8-part
> would be difficult without an ugly and perhaps run-free link section.
>
> I decided to abandon the method-based approach and simply look for an 8-part,
> hopefully an exact one. It turns out that there is still enough room for
> manoeuver in the graph containing all potential run-rows plus links between
> them involving at most 2 other rows. I investigated the possibilities, but
> the problem is that there is too much flexibility. An arbitrary 1/8th
> quarter peal length string of place notation is a lot to learn, and not very
> satisfying. I needed to impose more structure.
>
> One potential improvement is a 16-part, with the part ends being the cyclic
> ones 23456781 etc. plus their reverses (which form a dihedral group,
> isomorphic to plain hunt). I thought it might be possible to have a
> symmetric 8-part, with a forward part end part followed by a reverse part end
> one rung in reverse, but I can't see how to do this without asymmetric (and
> perhaps run-free) links between them, which isn't good.
>
> There is another way to get more symmetry - rotational symmetry. Suppose our
> part starts something like
> *12345678*
> 58 21435768
> x 12347586
> 16 13274568
> ...
> Then one can ring the same place notations reversed and in reverse order at
> the end of the part:
> ...
> 13452768
> 38 31425678
> x 13246587
> 14 *12345678*
> 58 21435768
> x 12347686
> 16 13274568
> ...
> With the chosen part ends (and no others), any potential run-row in one half
> will become a potential run-row in the other half, with runs at the front
> becoming into runs at the back and vice versa. And this will effectively
> turn the touch into a 32-part, with only half as much place notation to be
> learned and searched for.
>
> How does one avoid falseness in such a scheme? Every row has a 'partner',
> which is the one that we want to occur in the corresponding place in the
> other half of the part. E.g. the partner of 12345867 is 23145678. And
> taking partners respects the 16-part structure. (Mathematically the partner
> is the conjugate by reverse rounds R, and if P is a part end then R(PA)R =
> P'(RAR) where P' = RPR which is another part end since R is in the normalizer
> of the part end group).
>
> Here there is a delicate issue. Some rows are their own partner. Fortunately
> there are only two (modulo the 16 parts): rounds and 43215678. (In an 8-part
> it would be rounds and backrounds). Therefore we must use these as the
> apices. So what we want is a block starting from rounds and ending at
> 43215678 (or its equivalent in some other part), visiting either each other
> run-row or its partner (but not both). Then we reverse this block both in
> space and time to get back again and obtain one rotationally symmetric part.
> An inevitable consequence of this is that the part end after one part must be
> 43218765. On the face of it this is not ideal, because then there will be 8
> round blocks to be joined together somehow. But things will in fact turn out
> well...
>
> After some experimentation with different place notation selections and
> requiring a decent amount of traffic through 45 to avoid stagnation, I came
> up with block B, which I was pretty happy with. The different run types are
> nicely mixed, with often 2 or 3 immediate repetitions of the same type using
> dodges, but no more. Only good place notations appear (unless you object to
> 34 and 56, in which case I don't think there is much hope here). There are
> some other options, but not that many, particularly as short as this. As
> described above, the block is rotationally symmetric, and 8 copies of the
> round block BB give all the rows we want.
>
> That leaves the problem of joining the blocks. Initially I was not
> optimistic about doing this elegantly. The rather frenetic place notation
> means one should not expect that many Q-sets, especially without long places,
> and there is the danger of getting e.g. to the part end 34567812, so that one
> still has more joining to do.
>
> However, a miracle occurs. There is a magical Q-set present that involves
> all of the place notations 16,38,18,36 and preserves the rotational symmetry!
> Replacing the original 38 with 18 shunts to the corresponding place in the
> second half of the block, and we get back again by replacing 16 with 36, so
> only 2 blocks are joined, and we don't get a parity problem.
> 12348657 12348576
> 21384567 13284567
> becomes
> 12348657 12348576
> 13284567 21384567
> and we never need to interrupt the run-fest. Amazingly, the surrounding
> place notations are such that we (just) avoid long places. Furthermore, one
> shunt gets to 45678123, so no other link types are required. Many of us know
> that these things rarely work out so nicely!
>
> Incidentally, I don't think I have ever seen this type of Q-set used before.
>
> At this point it's easy: we just successively link the blocks with 7 such
> Q-sets. The whole touch has exact rotational symmetry. It is amusing that
> the A and C parts have odd length, so alternate part ends come at handstroke.
> I really find it amazing that all this works!
>
> What's the best way to describe this touch? As I have tried to argue, I
> think this should be entirely at the discretion of any band that wants to
> ring it. One-size-fits-all attempts to legislate simply get in the way of
> new ideas like this.
>
> However, if one wants to think in terms of methods, one suggestion (among
> other possibilities) is the following. It has the advantage of just two
> rotationally symmetric methods that don't look too daunting (a principle, and
> a short 11222 differential), and lead-end calls. The drawback is that the
> lead-end orders are unintuitive, and the cyclic structure gets rather
> severely hidden in this description.
>
> Frog Major:
> -4-34.6.5.6-4-4.56.1.56.4-34-3.6.5-56.4.36-4.56.1.34.1.34.5-34.
> 56-4.56.1.56.1.34.5-36.5.34-4.3.6-56-5.34.1.34.5-5-3.4.3.56-5-1
> l.e. 57628413
> bob = 3
>
> Toad Differential Major:
> 4-4-6.3.56.34.1.34-5.4-56.1.56.34.6.3-5-5.36
> l.e. 13254876
> bob = 6
>
> 12345678
> --------
> (F) 35426781
> F 68751234
> F 13284567
> F_T_ 64573218
> F_TT_ 31248765
> F_TT_ 86715432
> F_TT_ 53462187
> F_TT_ 28137654
> F_TT_ 75684321
> F_TT_ 42351876
> F_TT_ 17826543
> F_T_ 46537812
> F 71862345
> F 24315678
> F 57648123
> F 82173456
> (F) 12345678
> --------
> Start and finish at the half-lead of Frog: 56-4.56.1...
>
>
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