CHANGE RINGING IN MINNEAPOLIS/ST. PAUL

INTRODUCTION TO CHANGE RINGING

• the bells are first rung in order from highest to lowest pitch (rounds); thereafter
• every bell is struck once in each row;
• each bell moves at most one place from one row to the next;
• no permutation is repeated, until the last row returns to rounds.

• on 4 bells: 4! = 24
• on 5 bells: 5! = 120
• on 6 bells: 6! = 720
• on 7 bells: 7! = 5,040 Ringing 5000 or more changes counts as a "peal," which takes about 3 hours.
• on 8 bells: 8! = 40,320

For a sample of what change ringing sounds like, click on the bell:

(Saint Mark's bells, Philadelphia, ringing Grandsire Triples; thanks to Brian Zook)

Many more sound files are available at Recordings of Change Ringing

MATHEMATICAL INTERPRETATION I: SUBGROUPS AND COSETS

Possible operations on 4 bells, with ringing positions notated {A, B, C, D}, include:

• h: change the two outside pairs of bells; that is, apply the permutation (A,B)(C,D)
• b: change the inside pair of bells; that is, apply the permutation (B,C)
• g: change the last pair of bells; that is, apply the permutation (C,D).

Alternately applying h and b produces 8 different rows ("changes") before returning to rounds:
[see Budden, Table 24.04; the image shown here is similar]

"The scheme of eight changes is thus seen to be the group generated by b and h, or Gp (b, h). This cannot be the whole group S₄, for we know that S₄ requires three generators of period 2, and also that the group generated by two elements of period 2 is dihedral . . . . [I]t is more satisfactory to use h in conjunction with b (as we did in our preliminary attempt), but to get out of the group D₄ by the occasional use of g. . . Table 24.05 shows one method [Plain Bob Minimus] of getting all twenty-four changes on four bells. At row 8, to have used b would have produced rounds again [as before]. So we use g instead, and this enables us to get into a new set of eight permutations, {m, c, f, t, v, l, j, p}. Not unexpectedly, this is a coset of the subgroup H {1, h, k, w, y, r, n, b }, the reason emerging when we inspect the working in the final column. For example, row 13 (permutation v) is obtained from row 12 (permutation t) by applying the transposition b, so that bt = v. But t = wm (already obtained), so that v = bwm. But row 5 (= 13 - 8) shows that bw = y, hence v = ym, and so is in the right coset Hm. Similarly the final eight permutations belong to the coset Hi, the introduction of the permutation g at row 16 taking us across into this coset. Note that g was introduced each time the treble (No. 1) returned to the lead (position A)."
[see Budden, Table 24.05; permutation labels are as given in Budden, Table 18.04]

MATHEMATICAL INTERPRETATION II: CAYLEY GRAPHS AND HAMILTONIAN CYCLES

"The so-called b-bell Cayley graph has as its vertices the different b-bell changes, and two of its vertices are connected by an edge labeled by one of the (b-bell) transitions if and only if this transition is a transition between the two vertices. This means that this graph has b! vertices. The number of different labels of its edges-that is, the number of transitions in the symmetric group S_b-is F (b) - 1, where F (b) is the bth Fibonacci number . . . Remember that F(b) is defined recursively for any nonnegative integer b by setting F (0) = F (1) = 1, F (b + 1) = F (b) + F (b - 1). Hence, F (2) - 1 = 1, F (3) - 1 = 2, F (4) - 1 =4, F (5) - 1 = 7, and so on. There is exactly one edge of every kind ending at every vertex."
[see Polster, Fig. 6.3 "The 3-bell Cayley Graph," Fig. 6.4 "The 4-bell Cayley graph is a truncated octahedron with 'crosses across its square faces.'"]

"Note that if we do not label the vertices of this graph, none of these vertices is distinguished in any way among the rest; that is, the automorphism group of the graph acts transitively on the vertices. In fact, we do not lose any information about the graph by not labeling the vertices. To reconstruct a valid labeling of the vertices by changes, start by labeling an arbitrary vertex s (for "start") with an arbitrary change. Given any other vertex e (for "end"), choose a path in the Cayley graph that connects s with e. As you travel from s to e, successively apply the transformations that correspond to the edges that make up the path to turn the label for s into the lable for e. . . .Every cycle in the Cayley graph corresponds to two oriented cycles. Furthermore, the two ringing sequences corresponding to these two oriented cycles are inverses of each other."

[see Polster, Fig. 6.6 "Hamiltonian cycles in the 4-bell Cayley graph that correspond to the plain courses of the eleven methods on four bells. The common starting vertex is the upper-left vertex of the inner hexagon in Figure 6.4."]
"Remember that all these plain courses are 4-bell extents that have divisions into three leads of eight changes each. These order 3 symmetries of the extents translate into order 3 rotational symmetries of the corresponding cycles. The fact that all eleven methods are palindromic translates into order 2 (mirror) symmetries of the Hamiltonian cycles."

READING LISTS:

Dorothy Sayers' The Nine Tailors is an excellent mystery novel involving change-ringing.

Sources highlighted in the exhibit:

• FJ Budden. The Fascination of Groups. Cambridge: Cambridge University Press, 1972. Chapter 24: "Ringing the changes: groups and camapanology," pp 451-479.
  Mathematics Library 512.7 B859
• Burkard Polster. The Mathematics of Juggling. New York: Springer, 2003. Chapter 6: "Jingling, or ringing the changes," pp 141-176.
  Mathematics Library QA292 .P65 2003
• David Joyner. Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys. Baltimore: Johns Hopkins University Press, 2002. Section 3.5: "Permutations and bell ringing," pp 44-46; Section 5.12: "Campanology, revisited," p 96.
  Mathematics Library QA174.2 .J69 2002
• Ian Stewart. Another Fine Math You've Got Me Into--. New York: WH Freeman, 1992. Chapter 13: "The Group-theorist of Notre Dame," pp 199-219.
  Mathematics Library QA95 .S723 1992
• Kenneth J Falconer. "The Ancient English Art of Change-Ringing." In: Richard K Guy and Robert E Woodrow, editors. The Lighter Side of Mathematics. Washington DC: Mathematical Association of America, 1992: 261-264.
  Mathematics Library QA95 .L53x 1994

• A Dickinson. "Change-Ringing, Theory and Practice." Patterns of Symmetry, M. Senechal, G. Fleck, and A. Ludman, eds. Amherst: University of Massachusetts Press, 1977.
• DJ Dickinson. "On Fletcher's paper 'Campanological groups.'" Amer. Math. Monthly 64 (1957), 331-332.
• TJ Fletcher. "Campanological groups." Amer. Math. Monthly 63 (1956), 619-626.
• DG Papworth. "Computers and Change-Ringing." Computer J. , (1960) 47-50.
• BD Price. "Mathematical groups in campanology." Math. Gaz. 53 (1969), 129-133.
• RA Rankin. "A campanological problem in group theory." Math. Proc. Cambridge Philos. Soc. 44 (1948) 17-25.
• RA Rankin. "A campanological problem in group theory II." Math. Proc. Cambridge Philos. Soc. 62 (1966) 11-18.
• RG Swan. "A simple proof of Rankin's campanological theorem." Amer. Math. Soc., 160 (1999), 159-161.
• Arthur T White. "Graphs of groups on surfaces." Combinatorial surveys: Proceedings of the Sixth British Combinatorial Conference, ed. P.J. Cameron, Academic Press, London, 1977, 165-197.
• Arthur T White. "Ringing the changes." Math. Proc. Camb. Phil. Soc. 94 (1983), 203-215.
• Arthur T White. "Ringing the changes II." Ars Combinatorica 20A (1985), 65-75.
• Arthur T White. "Ringing the cosets." Amer. Math. Monthly 94 (1987), 721-746.
• Arthur T White. "A Hamiltonian construction in change ringing." Ars Combin. 25B (1988), 257-264.