# [r-t] Decisions / Algorithms for generating the extent

Richard Smith richard at ex-parrot.com
Thu Jun 22 16:53:13 UTC 2006

```I wrote:

>     1234     1243     1432
>   s 2134   s 2143     4123
>     2314     2413     4213
>     3241     4231     2431
>     3421     4321     2341
>     4312     3412     3214
>     4132     3142     3124
>     1423   s 1342   s 1324
>     ----     ----     ----
>     1243     1432     1234
>
> This touch seems quite reminiscent of Grandsire Doubles,
> with the fourth -- the pivot bell in the palindromic
> symmetry -- serving the role of observation.

Finding this made me go away and think a bit further about
extents of minimus.  Although Ander produced an exhaustive
list of minimus extents back in 2002, I hadn't looked at it
in enough details to see that there the 11 plain minimus
methods are not the only extents that have the treble plain
hunting throughout.

Ander's list is here

http://www.math.ubc.ca/~holroyd/minimus.html

and once you factor out rotations, reversals, reflections,
and "Kent/Oxford"-type variations (see Ander's webpage), the
list is reduced to just 29 extents.

http://www.math.ubc.ca/~holroyd/min29.txt

(We can define Kent/Oxford-type variants as methods that
differ only by sequences of four consecutive rows in one
method (a,b,c,d) appearing in the order (a,c,b,d) in the
same place in other.  On four bells there are three possible
"Kent/Oxford" variants: -12- == 34.12.34, -34- == 12.34.12,
and 12x12 == 34x34.)

These can be grouped by symmetry:

|  -   P   R  PGR |
---+----------------------
1 |  4  12   3   2  | 21
2 |  1   1   1      |  3
3 |      2       2  |  4
4 |      1          |  1
---+-----------------+----
|  5  16   4   4  | 29

- = Asymmetric
P = Palindromic symmetry
R = Rotational symmetry  [marked as D on Ander's website]
PGR = Full double symmetry (maximum possible)

Note that Mirror symmetry is not possible in extents of
minimus as there are only two changes (x and 14) with that
symmetry.  The only extents with only glide symmetry are
those formed by selectively choosing Kent/Oxford variations
to partially breaking the symmetry of fully symmetric (1PGR)
extent.

Clearly the 4P extent must be Erin / Stanton.  (Erin and
Reverse Stanton differ only by changing 34-34 to 12-12
around the six end.)  The 3PGR extents are Double Court and
Double Bob (and its variants: Double Canterbury, St Nicholas
and Reverse St Nicholas).  The 3P extents are Single Court
(and its variant, Reverse Court) and Plain Bob (and its
variants, Reverse Bob, Single Canterbury and Reverse
Canterbury).

Beyond this, the 24 one- and two-part extents are generally
ignored as uninteresting, however, they include some
interesting extents such as the one above.  In particular,
six extents have a hunting treble:

#7   1PGR    I|n|-|u|O
#14   1P--    I|-|n|-|-|-|n|I
#15   1P--    I|-|n|-|-|u|n|I
#16   1P--    I|-|n|n|-|u|n|I
#17   1P--    I|n|-|u|n|n|u|I
#23   1--R    O|-|n|-|-|-|n|O

The second of these (#14) is the one quoted at the start of
this email; the last (#23) is essentially the same touch,
but with one course of Original singled in a pair of 34
"singles"  and the other with a pair of 12 "singles" giving
the extent rotational rather than palindromic symmetry.

Extent #23 (with treble as hunt and fourth as "pivot")

1234     1243     1432
s 2134   s 2143     4123
2314     2413     4213
3241   s 2431   s 4231
3421     2341     4321
4312     3214     3412
4132     3124     3142
1423     1342     1324
----     ----     ----
1243     1432     1234

What of the other four extents?  These are just variantions
on the idea, but using more Q-sets.

Extent #7

1234     1432     1243
s 2134     4123   s 2143
2314     4213     2413
3241   s 4231   s 2431
3421     4321     2341
s 3412   s 4312     3214
3142     4132     3124
s 1342     1423   s 1324
----     ----     ----
1432     1243     1234

Extent #15

1234     1342     1423
s 2134   s 3142     4132
2314     3412     4312
s 2341     4321     3421
2431     4231     3241
4213     2413   s 3214
4123     2143     3124
1432   s 1243   s 1324
----     ----     ----
1342     1423     1324

Extent #16

1234     1243     1432
s 2134   s 2143   s 4132
2314     2413     4312
s 2341     4231     3421
2431     4321     3241
4213     3412   s 3214
4123     3142     3124
s 1423   s 1342   s 1324
----     ----     ----
1243     1432     1234

Extent #17

1234     1432     1243
s 2134   s 4132   s 2143
2314     4312     2413
3241   s 4321   s 2431
3421     4231     2341
s 3412   s 4213     3214
3142     4123     3124
s 1342   s 1423   s 1324
----     ----     ----
1432     1243     1234

With fixed treble, there are six possible Q-sets: they can
swap 23, 24 or 34 in the coursing order (by making those two
bells make adjacent places), and these can be done in two
ways depending on whether 12 or 34 singles are used.
Clearly to visit all three courses of Original, at least two
singles involving all three bells are needed.  After that
most (though not all) combinations of Q-sets produce an
extent.

Q-sets        |               |        |
34        12     |   Sizes of    | Extent | Symmetry
--------  --------  |  Partitions   | Number |
23 24 34  23 24 34  |               |        |
---------------------+---------------+--------+----------
|   8 + 8 + 8   |    -   |   3PGR
X                   |  16 + 8       |    -   |   1P--
X  X             |  24           |   14   |   1P--
X         X         |   8 + 8 + 8   |    -   |   1PGR
X      X      |  24           |   23   |   1--R
X  X  X             |  24           |    4   |   3P--
X  X         X   |  16 + 8       |    -   |   1---
X  X   X         |  24           |   15   |   1P--
X  X      X  X   |  24           |    7   |   1PGR
X  X   X     X   |  16 + 8       |    -   |   1---
X  X  X   X         |  24           |   16   |   1P--
X  X  X      X  X   |  24           |   17   |   1P--
X  X  X   X  X  X   |  24           |    1   |   3PGR

(NB: The Q-set combinations listed have selected so that the
4th is pivot bell where there is enough symmetry for there
to be a pivot bell.  The missing combinations are simply
rotations or reflections of the ones listed.)

Interestingly, other than the three choices of Q-sets that
clearly cannot work as they do not visit each course (lines
1, 2 and 4 in the table), the remaining choices of Q-sets
that do not produce an extent are precisely those choices
that have no symmetry.  This is perhaps unsurprising at one
level -- ringing problems often seem to be biased towards
symmetrical solutions -- but is there a "good" reason why
this should be the case?

RAS

```