Mathematics of bell-ringing Dr Rob Sturman University of Leeds, School of Mathematics 24th September 2016 Gravity Fields Festival Easy example: on 3 bells, the possible sequences are: 1 2 3 9 > 1 3 2 > > Combinatorics: 2 1 3 => 3 choices for first place, then 2 6 sequences 2 3 1 choices, then 1 choice =) > 3 1 2 > 3 × 2 × 1 = 3! (3 factorial) > 3 2 1 ; 4 bells: 4! = 4 × 3 × 2 × 1 = 24 sequences (around 30 seconds) 6 bells: 6! = 720 sequences (takes about 25 minutes) 8 bells: 8! = 40320 sequences (18 hours, in Loughborough, 1963) 12 bells: 12! = 479; 001; 600 sequences (over 30 years!) Change-ringing Extents Basic idea: Ring every possible sequence exactly once 4 bells: 4! = 4 × 3 × 2 × 1 = 24 sequences (around 30 seconds) 6 bells: 6! = 720 sequences (takes about 25 minutes) 8 bells: 8! = 40320 sequences (18 hours, in Loughborough, 1963) 12 bells: 12! = 479; 001; 600 sequences (over 30 years!) Change-ringing Extents Basic idea: Ring every possible sequence exactly once Easy example: on 3 bells, the possible sequences are: 1 2 3 9 > 1 3 2 > > Combinatorics: 2 1 3 => 3 choices for first place, then 2 6 sequences 2 3 1 choices, then 1 choice =) > 3 1 2 > 3 × 2 × 1 = 3! (3 factorial) > 3 2 1 ; 12 bells: 12! = 479; 001; 600 sequences (over 30 years!) Change-ringing Extents Basic idea: Ring every possible sequence exactly once Easy example: on 3 bells, the possible sequences are: 1 2 3 9 > 1 3 2 > > Combinatorics: 2 1 3 => 3 choices for first place, then 2 6 sequences 2 3 1 choices, then 1 choice =) > 3 1 2 > 3 × 2 × 1 = 3! (3 factorial) > 3 2 1 ; 4 bells: 4! = 4 × 3 × 2 × 1 = 24 sequences (around 30 seconds) 6 bells: 6! = 720 sequences (takes about 25 minutes) 8 bells: 8! = 40320 sequences (18 hours, in Loughborough, 1963) Change-ringing Extents Basic idea: Ring every possible sequence exactly once Easy example: on 3 bells, the possible sequences are: 1 2 3 9 > 1 3 2 > > Combinatorics: 2 1 3 => 3 choices for first place, then 2 6 sequences 2 3 1 choices, then 1 choice =) > 3 1 2 > 3 × 2 × 1 = 3! (3 factorial) > 3 2 1 ; 4 bells: 4! = 4 × 3 × 2 × 1 = 24 sequences (around 30 seconds) 6 bells: 6! = 720 sequences (takes about 25 minutes) 8 bells: 8! = 40320 sequences (18 hours, in Loughborough, 1963) 12 bells: 12! = 479; 001; 600 sequences (over 30 years!) You can only swap neighbouring bells. Easy example: 3 bells: a = (12), b = (23) [positions, not bells] Alternate these changes: All 6 possible sequences gener- ated, and the ringers only have 1 2 3 9 to remember two different opera- > a 2 1 3 > tions, and to alternate them. > > b 2 3 1 => a 3 2 1 6 sequences b b 3 1 2 > > a 1 3 2 > 1 > a b 1 2 3 ; 3 2 Change-ringing A constraint and an opportunity You can’t just move from one sequence to any other! Easy example: 3 bells: a = (12), b = (23) [positions, not bells] Alternate these changes: All 6 possible sequences gener- ated, and the ringers only have 1 2 3 9 to remember two different opera- > a 2 1 3 > tions, and to alternate them. > > b 2 3 1 => a 3 2 1 6 sequences b b 3 1 2 > > a 1 3 2 > 1 > a b 1 2 3 ; 3 2 Change-ringing A constraint and an opportunity You can’t just move from one sequence to any other! You can only swap neighbouring bells. All 6 possible sequences gener- ated, and the ringers only have to remember two different opera- tions, and to alternate them. b 1 a 3 2 Change-ringing A constraint and an opportunity You can’t just move from one sequence to any other! You can only swap neighbouring bells. Easy example: 3 bells: a = (12), b = (23) [positions, not bells] Alternate these changes: 1 2 3 9 > a 2 1 3 > > > b 2 3 1 => a 3 2 1 6 sequences b 3 1 2 > > a 1 3 2 > > b 1 2 3 ; Change-ringing A constraint and an opportunity You can’t just move from one sequence to any other! You can only swap neighbouring bells. Easy example: 3 bells: a = (12), b = (23) [positions, not bells] Alternate these changes: All 6 possible sequences gener- ated, and the ringers only have 1 2 3 9 to remember two different opera- > a 2 1 3 > tions, and to alternate them. > > b 2 3 1 => a 3 2 1 6 sequences b b 3 1 2 > > a 1 3 2 > 1 > a b 1 2 3 ; 3 2 9 The full peal is not generated > by a and b — we need an extra > > change, for example > > > c = (34). => only 8 rows! a > > 1 2 > > > > ;> 3 4 b Change-ringing Plain Bob Minimus Consider 4 bells and let a = (12)(34) , b = (23) 1 2 3 4 a 2 1 4 3 b 2 4 1 3 a 4 2 3 1 b 4 3 2 1 a 3 4 1 2 b 3 1 4 2 a 1 3 2 4 b 1 2 3 4 The full peal is not generated by a and b — we need an extra change, for example c = (34). a 1 2 3 4 b Change-ringing Plain Bob Minimus Consider 4 bells and let a = (12)(34) , b = (23) 1 2 3 4 9 > a 2 1 4 3 > > b 2 4 1 3 > > > a 4 2 3 1 => b 4 3 2 1 only 8 rows! a 3 4 1 2 > > b 3 1 4 2 > > a 1 3 2 4 > > b 1 2 3 4 ; Change-ringing Plain Bob Minimus Consider 4 bells and let a = (12)(34) , b = (23) The full peal is not generated 1 2 3 4 9 > by a and b — we need an extra a 2 1 4 3 > > change, for example b 2 4 1 3 > > > c = (34). a 4 2 3 1 => b 4 3 2 1 only 8 rows! a a 3 4 1 2 > 1 2 > b 3 1 4 2 > > a 1 3 2 4 > > b 1 2 3 4 ; 3 4 b Change-ringing Plain Bob Minimus — full extent a = (12)(34) , b = (23), c = (34) 1 2 3 4 c 1 3 4 2 c 1 4 2 3 a 2 1 4 3 a 3 1 2 4 a 4 1 3 2 b 2 4 1 3 b 3 2 1 4 b 4 3 1 2 a 4 2 3 1 a 2 3 4 1 a 3 4 2 1 b 4 3 2 1 b 2 4 3 1 b 3 2 4 1 a 3 4 1 2 a 4 2 1 3 a 2 3 1 4 b 3 1 4 2 b 4 1 2 3 b 2 1 3 4 a 1 3 2 4 a 1 4 3 2 a 1 2 4 3 c 1 2 3 4 Change-ringing Plain Bob Minor — full peal Change-ringing Extent rules 1 start and end with bells in order musical satisfaction 2 every sequence rung exactly once completeness 3 only swap neighbouring bells physical contraints 4 each bell must move at least every other change keep ringers interested 5 bells do the same work as each other keep ringers equitable 6 the method is palindromic ease of memory? Fabian Stedman 1640–1713 Fabian Stedman: “the father of change- ringing” a contemporary of Isaac Newton (1643–1727) Born in Herefordshire, went to London as apprentice printer in 1655 Learnt bell-ringing at St Mary-le-Bow two publications of bell-ringing a parish clerk at St Bene’t’s Church in Cambridge, 1670 cross-changes Stedman Doubles Stedman Doubles a = (12)(45), b = (23)(45) 1 2 3 4 5 1 2 3 4 5 a 2 1 3 5 4 b 1 3 2 5 4 b 2 3 1 4 5 a 3 1 2 4 5 a 3 2 1 5 4 b 3 2 1 5 4 b 3 1 2 4 5 a 2 3 1 4 5 a 1 3 2 5 4 b 2 1 3 5 4 b 1 2 3 4 5 a 1 2 3 4 5 Stedman Doubles a = (12)(45), b = (23)(45), c = (12)(34) (a parting change) 1 2 3 4 5 3 1 5 2 4 a 2 1 3 5 4 b 3 5 1 4 2 b 2 3 1 4 5 a 5 3 1 2 4 a 3 2 1 5 4 b 5 1 3 4 2 b 3 1 2 4 5 a 1 5 3 2 4 a 1 3 2 5 4 b 1 3 5 4 2 c 3 1 5 2 4 c 3 1 4 5 2 Now known as a single, d = (34) will then allow the other sixty combinations to be rung. Grandsire doubles on 5 bells: a = (12)(45), b = (23)(45), c = (12)(34) =) a(bc)4a Always an even number of permutations, so only half the 5! = 120 possible sequences generated. Need a ‘single’, e.g., d = (45) to generate the other ‘odd’ permutations. Stedman Doubles a = (12)(45), b = (23)(45), c = (12)(34) (a parting change) So the ‘lead’ (ababac)(bababc) = (ab)2ac(ba)2bc gives the permutations 12345 ! 31452 ! 43521 ! 54213 ! 25134 ! 12345 “By this method the peal will go to sixty changes, and to carry it farther, extremes must be made”.