Maths and ringing

This is an overview of a talk I gave to maths pupils at Park House School, Newbury , together with references for further study following the talk. I later gave a shortened version of the talk to pupils at Sir William Borlase's Grammar School, Marlow.

Read the article about the talk that was published in The Ringing World.

 If you would like a similar talk, please contact me .

Overview

The talk explains the mathematical underpinning of change ringing, and in the process introduces many mathematical concepts that have more widespread applicability, including: permutations, factorials, symbolic transformations, groups and group theory, Hamiltonian circuits, symmetry and the Fibonacci series.

• Introduction – Mathematical music Background & history. Constraints on bell music. – Changes Transformations to generate one row from another

• Exercise – Rows and changes Find how many of each as function of N. [Worksheet provided] – Explanation of answers 

• Basic change ringing – Interesting tunes Description of some musical rows, illustrated with sound – Relationships between rows Sequences of changes used to generate specific rows

• Exercise – Repeating the Queens transform Explore effect of repeating the process on different numbers of bells [Worksheet provided] – Explanation of answers

• Methods – All change Description of hunting – Going further Construction & properties of a basic method – Touches and extents Construction of longer pieces of ringing

• Exercise – Compose a quarter peal of Plain Bob Royal Pupils suggest where to put calls, observe the effect, discuss what to do next, etc. [Whole group exercise, supported by software]

• More complexity – Method structure Structural relationships between members of method families, and how they affect the paths of bells – Truth and falseness Explanation of the need for ‘truth’ and of the problem of internal falseness – Bobs only Stedman Triples The story of ringing’s equivalent of the 4-colour map theorem

• A different way of looking – Spatial models Describe the mapping of rows to spatial models for N=3, 4, 5. Show the models. – Related work Summarise the contents of some even earlier work with a tantalising end. – Other approaches Describe how many people have mapped leads of methods to spatial models.

• Music – Musical effects on higher numbers Demonstrate progression from mainly sequential to mainly alternating music on 12.

• Exercise – Walk plain hunt on 8 Pupils, each with a bell, swap places while ringing, to emulate the effect of changing.

• Conclusion – Wind up Point to list of references. Anecdote about ringing on TV. – Questions .

See also: my general Talk on the Maths of Bellringing . See also: details of other Lectures for secondary school pupils  See also: Music and Ringing 

References

1 – Stedman, Fabian, “Campanologia”, 1677

2 – Griffiths, Deryn, “Twin Bob Plan Compositions of Stedman Triples: Partitioning of graphs into Hamiltonian subgraphs as used in bellringing”, Nov 1994 http://maths.usyd.edu.au/res/CompAlg/Gri/2bob-sted.ps  

3 – Johnson, Andrew, “Bobs-only Stedman Triples made easy”, The Ringing World, 11 August 1995, p841

4 – Price, Brian D, “More about Q-Set Laws”, The Ringing World, 3 April 1953

5 – “QED”, “A Note upon Some Aspects of Stedman Triples”, The Ringing World, 20 February 1953 also http://myweb.tiscali.co.uk/saddleton/stedman/qed.htm 

6 – Saddleton, Philip and Johnson, Andrew, “Peals of Stedman Triples with bobs only”, 1995, – http://myweb.tiscali.co.uk/saddleton/stedman/bdp.htm 

7 – “Software Firsts” http://www.ringing.info/shaggy.html 

8 – King, Frank H, “ Equations of Motion of a Free Bell and Clapper” http://www.cl.cam.ac.uk/~fhk1/equations.pdf 

9 – White, Arthur T, “Graphs, groups and surfaces”, 1973

10 – White, Arthur & Wilson, Robin, “The Hunting Group” , The Mathematical Gazette, Vol. 79, No. 484. (Mar., 1995), pp. 5-16. http://links.jstor.org/stable/3619985

11 – White, Arthur T, “Fabian Stedman: The First Group Theorist?”, The American Mathematical Monthly, Vol. 103, No. 9 (Nov., 1996), pp. 771-778 http://links.jstor.org/stable/2974446  

12 – Leigh Silver, A. L. “Some Musico-Mathematical Curiosities” The Mathematical Gazette, Vol. 48, No. 363 (Feb., 1964), pp. 1-17. http://www.jstor.org/stable/3614300 

13 – Roaf, Dermot & White, Arthur, “Ringing the changes: bells & mathematics” in “Music and Mathematics”, 2003 12 – Fortran Friends, (polyhedra to download, etc) http://fortran.orpheusweb.co.uk/Bells/polyhedra/ 

14 – Pumphrey, Hugh, “Bell-ringing methods as polyhedra”, http://xweb.geos.ed.ac.uk/~hcp/bells/  

15 – Burkard Polster and Marty Ross, “Ringing the Changes”, http://plus.maths.org/issue53/features/polsteross/ 

16 – Harrison, John “Ringing shapes” http://jaharrison.me.uk/Ringing/RingingShapes/  

17 – Harrison, John, Downloadable Minimus model ” http://jaharrison.me.uk/Ringing/RingingShapes/Download/Minimus.pdf  

18 – Harrison, John “ Bellringing and Music” http://jaharrison.me.uk/Ringing/Music/ 

19 – Harrison, John “Murder in the belfry” http://jaharrison.me.uk/New/Articles/RingOutDead.html  

20 – More links on ringing and maths – http://www.ringing.info/  Look under ‘Compositions’ -> Group Theory

21 – YouTube video, Change ringing on handbells with the 'cross & stretch' style – http://www.youtube.com/watch?v=yRy7YmUS3uQ 

22 – YouTube video, Change ringing on 24 handbells (including mega-tittums) at The Ringing World Centenary – http://www.youtube.com/watch?v=4-fCRBNTNp0 

23 – 'Mobel', mobile bellringing simulator for i-Phone – http://abelsim.co.uk/MobelNews.htm