The art or “exercise” of change ringing is a kind of mathematical team sport dating from the 1600s. It originated in England but now is found all over the world. A band of ringers plays long sequences of permutations on a set of peal bells. Understanding the patterns so they can be played quickly from memory is an exact mental exercise which takes months for ringers to perfect. Composers of new sequences must understand the combinatorics of permutations, the physical constraints of heavy bells, and the long history of the art and its specialized vocabulary. Change ringing is a little-known but surprisingly rich and beautiful acoustical application of mathematics.

The video shows members of the Washington Ringing Society of Washington, D.C., ringing a long touch of Grandsire Caters.

Related:

More videos from the Mathematical Impressions series.

The method (plain bob on 4 bells) described in the middle of the video can be viewed as an enumeration of the 24 elements of the symmetric group S_4 by enumerating the elements of a dihedral subgroup of order 8 and its cosets (where the permutation that switches the bells in positions 3 and 4 at the bottom of each column switches you from one coset to the next). There is a nice article which discusses this use of cosets in bell ringing:

Arthur White, “Ringing the Cosets”, The American Mathematical Monthly, Vol. 94, No. 8 (1987), pg 721–746.

For anyone interested in learning how to ring tower bells, more information (including a tool for locating towers near you) can be found at the North American Guild of Change Ringers website: http://www.nagcr.org

Great videos. We have referred to them a few times in our blog on math blogs at the AMS. Please continue!

There are interesting mappings between change ringing and spatial models, see: http://jaharrison.me.uk/Ringing/RingingShapes/

Anyone interested in music might be interested to see how it is shaped by the mathematical constraints, see: http://jaharrison.me.uk/Ringing/Music/

Learning to ring properly takes years, not months. Once the basics have been mastered, learning a new “method” (pattern) may not take an experienced ringer very long.