Happy Holidays Math 152!
A few Saturdays ago, Kate took a couple of us to visit the change ringing rehearsal at Old North Church in Boston. Old North Church houses the oldest bells in North America, which have hung in the church spire since 1745. There are 8 bells, and they are huge! They weigh between 500 and 1300 pounds.
Change ringing and group theory are related because the mechanics of the bells limits the ringers’ ability to control them once they are ringing. In fact, the only changes in the order of pitches heard come from two bells switching places in the order, which is like the transpositions we studied in class. In change ringing, the goal is to ring a peal, or a series of permutations of the pitches in which no permutation is permitted. A full peal is such a series when all the possible permutations of pitches are rung. We learned that such a peal can take a very long time, depending on the number of the bells! Since there are n! possible permutations of n bells, ringing the 40,320 permutations of 8 bells could take almost an entire 24-hour day. A full peal on 6 bells is far more manageable–it only takes 20 or 30 minutes.
Ringers typically memorize methods, patterns of ringing that allow the ringers to go through the permutations without memorizing each individual one. We learned about a simple pattern. You start with bells 1, 2, 3, 4 in that order. Then, you alternate between switching the order of the interior two bells and switching the order of the pairs of outside bells. Sorry if this doesn’t make any sense–let me show you:
1 2 3 4
1 3 2 4 (switch interior pair)
3 1 4 2 (switch LHS and RHS pairs)
3 4 1 2 (switch interior pair)
4 3 2 1
4 2 3 1
2 4 1 3
2 1 4 3
1 2 3 4
(You repeat the 1234 pattern at the end.) Another cool peal we learned about was a full peal of 5 bells, where the 60 even permutations are rung, a transposition (as in bell switch) is called, and the 60 odd transpositions are rung to make up the full peal.
We had lots of fun visiting the change ringers at Old North and even got to take in the view at the top of the spire! If you are interested in learning to ring bells, they urge that you visit them during their rehearsals from 11 to 1 on Saturdays because they are always looking for new people to learn!
Jack learning how to ring:
“One if by land, two if by sea”: