Church bells are huge things. A
typical heaviest bell in a church tower weighs close to a ton, and quite a bit
of practice is needed before you can handle it safely. Church bells are
attached to wheels, and the whole assembly is made to rotate by pulling on the
rope wound round the wheel.
One consequence is that it takes about two seconds between consecutive strokes of a single bell, so that you can’t play tunes on church bells. Bellringers attempt instead to ring all the possible “changes” on a collection of bells. A change is the ringing of each bell, one after another, once each and without repeats; if there are 6 bells, numbered 1 to 6 (1 is highest, unlike musical terminology), a typical change might be 2 1 5 3 6 4, and such a change takes about two seconds to ring.
The Mathematics
The number of possible changes
is, naturally, the factorial of the number of bells:
5 120 4 minutes
6 720 24 minutes
7 5040 3 hours
8 40320 24 hours …
A complete set of changes (an
“extent”) on 8 bells has been rung only twice.
The bell-ringers’ problem is how
to ring all n! changes, without repeating any change, and subject to
these rules:
1. You cannot alter the
position of any bell from one change to the next by more than 1 place. (This is
because the weight of the bell prevents it; the natural period of oscillation
of a bell is almost fixed.) Thus you can follow
2 1 5 3 6 4 by 1
2 3 5 4 6 but not by, for
example, 1 2 3 4 5 6, as both bells 4 and 5 would have to move by 2 places.
2. Each bell must follow the
same sequence (but starting in different places). This is largely to help ringers
to memorise the “method”.
There are literally thousands of ways that have been worked out to do this. They are given splendid names, such as Grandsire Doubles (5 bells), Cambridge Surprise Minor (6 bells), Stedman Triples (7 bells) or Kent Treble Bob Major (8 bells). (The last word in the name tells you how many bells are involved.)
Let us see a simple method of trying to ring all 120 changes on 5 bells.
There are two absolutely
fundamental permutations, which I’m going to call f and g:
f: 1 2
3 4 5 ® 2
1 4 3 5 [in cycle notation this is f =
(1 2) (3 4)]
g: 1
2 3 4
5 ® 1
3 2 5 4 [g = (2 3) (4
5)]
So, if r is the order 1 2
3 4 5 (the basic bell-ringer’s order, called “rounds”), then
f(r) = 2 1
4 3 5
and gf(r) = 2 4
1 5 3
If you continue to do f followed by g, you will do 10 different permutations, but get back to rounds. These 10 permutations form a “subgroup” of the 120 permutations of 5 objects, denoted here by H. This subgroup is called the “hunting subgroup”; if you write out the table you will see that each bell moves by 1 places at a time, between the first and last places in the row, pausing only at the first and last places to ring twice in that position. This is a basic manoeuvre that bell-ringers learn, and it’s the foundation of most methods.
How to get the other 110 changes?
An obvious thing to do is to
introduce a different change instead of the last, fifth, g that would produce
rounds. If you replace that last g by
m: 1
2 3 4
5 ® 1
2 4 3 5 [m = (3 4)]
you get the change that is
characteristic of the method known as Plain Bob.
If you do change m every 10th change, and the rest of the time alternate between f and g, you will get 40 different changes before getting back into rounds. Changes 10–19 form what is called a “coset” of H; changes 20-19 and 30-30 form two more cosets. (One of the great theorems of group theory is that cosets always have the same number of different elements as the subgroup from which they are derived, in this case 10.)
Finally, to
get the remaining 80 changes, you need to put in a fourth different change,
which for this method will be
b: 1 2
3 4 5 ® 1
3 2 4 5 [b = (2 3)]
on the 9th, 49th and 89th changes (or the 19th, 59th and 99th, or the 29th, 69th and 109th changes). To get the ringers to do this, the leader of the ringers calls “Bob!” at the appropriate moments. The call is nothing to do with anyone named Robert; it’s just traditional. This will produce a further collection of cosets, each of 10 changes, and after the last of them the bells come back into rounds again, having achieved all 120 changes. These ideas can be applied to larger numbers of bells.
The History
And now for a remarkable piece of
mathematical history. The theory of groups was developed around 1820–30 by two
utterly brilliant and tragically short-lived mathematicians, the Norwegian
Niels Abel (1802–1829) and the Frenchman Évariste Galois (1811–1832). Among
their work was the proof that it is impossible to find a general formula to
solve a quintic equation, so that, for example, the solutions to the equation x5
– 6x + 3 = 0 cannot be written exactly in terms of nth roots for
any integer n. (An amazing result – and proved when Galois was only 19.)
Group theory has since been used for an extraordinary range of applications,
including error-correcting codes for getting messages back from space probes
and encrypting CDs so that they will play even when damaged. But all the group
theory needed for bell-ringing had been developed by Fabian Stedman in,
unbelievably, 1668.
Many mathematicians are bell-ringers. On the other hand, most bell-ringers would deny that they are in any way mathematical, although in fact they are doing quite complicated mathematics – without realising it – every time they ring.
If you are
interested, even if it’s only to visit a tower and watch bells being rung,
Google either “Middlesex” or “Surrey” “Association of Church Bell Ringers”.
Meanwhile, Dorothy L. Sayers wrote a classic crime novel, The Nine Tailors,
that has bell-ringing as its central theme. Set in an East Anglian village
community, it is also a magnificent novel in its own right.