[r-t] Decisions / Algorithms for generating the extent
Richard Smith
richard at ex-parrot.com
Mon Jun 26 14:00:14 UTC 2006
I wrote:
> Another way is to use Plain Changes on (n-r) bells as the
> coursing order for a singles-only touch of an n-bell method
> with r fixed bells which get into each position relative to
> each other in the plain course.
Now that I've thought about this a little further, this is
rather more subtle than I first thought. When we apply
Plain Changes on (n-1) to the coursing order of Original on
n, we not only visit every coursing order, but also every
coursing order backwards. As these two courses are false
against each other, what prevents this from making the touch
false? And what, for that matter, prevents part of the
course from being omitted?
In Plain Changes on three or more bells, the pair of bells
swapped to reach any given row is different from the pair
swapped to leave that row. This means that when applied to
the coursing order of Original, we enter and leave the
course in different places -- so, assuming only a single
type of single is used in the Original, we necessarily omit
part of the course.
Quoting from my original example of Plain Changes on 3
starting from the coursing order 243:
> 243 swap 3 & 4
> 234 swap 2 & 3
> 324 ...
This generates the following fragment of Original:
> 1234
> s 2134 < enter course 234 by swapping 3 & 4 in c.o.
> 2314
> 3241
> 3421
> 4312
> 4132
> s 1432 < leave course 234 by swapping 2 & 3 in c.o.
> ----
> 1342
I.e. we've only had six of the eight rows from the 234
course. The missing section is the bit around the lead end:
1423
1243
This has to be reached and left via a pair of singles
(otherwise it's false against the fragment above), and it
has to be rung backwards so that the coursing order is 432
(otherwise we have the same coursing order twice, which
would imply Plain Changes is false and we know that it
isn't.) So:
2143
s 1243 enter course 432 by swapping 3 & 4 in c.o.
----
1423
s 4123 leave course 432 by swapping 2 & 3 in c.o.
Clearly both fragments have to be entered and left by
swapping the same pairs of bells in coursing order --
otherwise they would either overlap or omit part of the
course.
If we look at the preceding and superseding coursing orders
for the two fragments, we see that they are also reverses of
each other, and that the (plain) changes that join them are
also reverses of each other.
Fragment #1 Fragment #2
2 4 3 3 4 2
| \/ \/ |
| /\ /\ |
2 3 4 4 3 2
\/ | | \/
/\ | | /\
3 2 4 4 2 3
We can now apply the same logic to the adjacent coursing
orders, and get eventually get two mutually-reverse sections
that join together to form a glide symmetric round block.
As Plain Changes are glide symmetric (in fact, they are
maximally symmetric -- i.e. they have palindromic, glide and
rotational symmetry), they can be used on the coursing order
of Original to produce an extent.
More generally, any glide symmetric extent of single changes
on (n-1) bells can be used to produce an extent of Original
on n bells, and no extent of singles changes on (n-1)
bells without glide symmetry can produce an extent of
Original on n bells.
>From Ander's list, we can see that there are only four
extents of minimus that only use single changes:
#1 3PGR I|n|O (Double Court)
#2 3PGR In=|O (Double Canterbury)
#9 2--R O|n|n|u|O
#22 1P-- In=|u|=n=|n|u|I
(See http://www.math.ubc.ca/~holroyd/min29.txt for notaton.)
Double Court and Double Canterbury are variants of Plain
Changes using near and far extremes respectively. As an
example of these, we can generate a fixed-treble extent of
Original Doubles using 345 singles in place of 5. As 34
(the bells swapped by the single) are not adjacent in the
standard coursing order, we need to choose an alternative
coursing order by choosing every other bell from the usual
one (exactly as for the extent of Bob Minor in the previous
email). This gives a starting coursing order of 4325 from
which we can ring Double Court with the 5 as the hunt bell:
4325 2435 3245
---- ---- ----
4235 2345 3425
4253 2354 3452
4523 2534 3542
5423 5234 5342
5243 5324 5432
2543 3524 4532
2453 3254 4352
2435 3245 4325
This gives a three part extent of doubles through which the
treble plain hunts.
12345 15243 12543 14253
21435 51423 s 21543 s 41253
24153 54132 25134 42135
42513 45312 52314 24315
45231 43521 53241 23451
s 54231 s 34521 s 35241 s 32451
52413 35412 32514 34215
25143 53142 23154 43125
21534 51324 21345 41352
s 12534 15234 12435 s 14352
----- ----- ----- -----
15243 12543 14253 13425
Twice repeated
But if the same is tried with a non-glide symmetric extent,
it fails. For example, extent #9 has rotational, but not
the required glide symmetry:
4325 2543
3425 5243
3245 5423
3254 5432
3524 5342
3542 5324
3452 5234
4352 2534
4532 2354
4523 2345
4253 2435
2453 4235
---- ----
2543 4325
Attempting to fit this to Original only lasts 39 rows before
becoming false.
12345 15234* 14532 15432
s 21345 51324 41352 s 51432
23154 53142 43125 54123
32514 35412 34215 45213
35241 34521 32451 42531
s 53241 s 43521 s 23451 s 24531 False rows marked
52314 45312 24315 25413 with a *.
25134 54132 42135 52143
21543 51423 41253 51234
s 12543 s 15423 14523 s 15234*
----- ----- -----
15234 14532 15432
The idea of apply Plain Changes to the coursing order is
given in Knuth's "The Art of Computer Programming"
<http://www.ex-parrot.com/~richard/papers/fasc2b.ps>.
as Exercise 69 on page 32, and attributed (page 48) to E.S.
Rapaport [Scripta Math. 24 (1959), pp51-8]. I've not
managed to lay my hands on a copy of this paper, though it
seems to be cited quite frequently. (Interestingly,
Rapaport is quoted as being "particularly motivated by bell
ringing". Does anyone know anything about this person?
Were/are they a ringer?)
RAS
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