A change-ringing tower has N tuned bells. Each one swings on a wheel; a ringer pulls a rope. Between strikes a bell takes about two seconds to come around. The bell weighs hundreds of kilograms and has its own inertia, so you cannot reorder it freely. A bell currently in position 3 of the sequence can move to position 2 or 4 next time, or stay — but it cannot jump to position 1. The physical constraint is the rule. The group theory follows.
So the constraints are:
· each of the N bells strikes exactly once per row
· between rows, each bell moves at most one position
· no row may repeat (you'd be wasting bells)
A method is a memorized rule that generates rows from previous rows mechanically. No caller needed in the simplest case — the rule is the score.
Minimus = 4 bells. 4! = 24 distinct rows. Plain bob minimus has the elegant property that one plain course is exactly 24 rows: it visits every permutation and returns to rounds without any calls. The method itself produces the full extent.
Place notation: x14x14,12
· x — every position changes; bells in (1,2) swap and bells in (3,4) swap simultaneously
· 14 — positions 1 and 4 stay put; positions 2 and 3 swap
· 12 — positions 1 and 2 stay put; positions 3 and 4 swap
Each lead is 8 changes: x 14 x 14 x 14 x 12. The comma marks the half-lead by palindromic symmetry. Three leads make a course.
Derived by hand from the place notation:
rounds: 1234 x 2143 14 2413 x 4231 14 4321 x 3412 14 3142 x 1324 12 1342 ← lead end 1 x 3124 14 3214 x 2341 14 2431 x 4213 14 4123 x 1432 12 1423 ← lead end 2 x 4132 14 4312 x 3421 14 3241 x 2314 14 2134 x 1243 12 1234 ← back to rounds
Bell 1 (the treble) over the first lead: position 1→2→3→4→4→3→2→1→1. It walks to the back and walks home. This is called plain hunting. All bells plain-hunt by default; the 14 and 12 places are what makes it bob rather than hunt — they deflect one or two bells off the pure hunt path and generate new permutations.
The plain course is exactly the extent on four bells. On five bells the plain course is 40 rows out of 120 possible. On six bells it's 60 out of 720. The plain course is a closed orbit; it does not reach every permutation for N > 4.
To reach the rest, conductors call bobs and singles at lead-ends — momentary deviations from the method. A bob says: instead of the usual lead-end place, do this other place. That small substitution rotates you into a different orbit. Stack them right and you visit every permutation exactly once. That's a peal: 5040 rows on six bells, about three hours, no row repeated.
The rule is a property of the substrate, not a design choice. No one decided bells should move adjacent-only. They decided because the bell weighs 800kg and swings on a wheel and you cannot rush it. The math became inevitable from the physics. Group theory grew up around campanology in the seventeenth century, before anyone called it group theory; Tintinnalogia was 1668. The mathematicians arrived later and found a vocabulary already in the towers.
A method has a name. Plain bob, grandsire, stedman, kent treble bob. Each is a different rule. Ringers learn them by the shape of the treble's path through the row — they call it the blue line. You memorize the curve of one bell and infer your own position from it.
· why palindromic place notation guarantees you return to rounds (the symmetry must force it; i haven't worked through why)
· what spliced means — combining methods mid-touch
· the social structure of a band: how eight people coordinate without one of them visibly conducting
These are the questions a next session can pick up.