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    What’s change ringing? I’ve never heard of it.

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      it’s a bell (think church bells) ringing technique surprisingly closely related to group theory in mathematics. Here’s a video explaining theory and practice: https://m.youtube.com/watch?v=f5GmUxl2NaU

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        tl;dr:

        • Ringing church bells (or handbells);
        • every permutation is different;
        • bells don’t move more than one place from one permutation to the next (think braiding);
        • you need a pattern of changes that produces the sequence of permutations

        Take a church tower with usually six bells, but let’s say four, to keep the examples simple.

        Now ring those bells in order from high to low:

        1 2 3 4
        

        Next ring the bells in a different order (a different permutation), but each bell may only (a) swap places with the bell before or after it, or (b) stay in place. For example, you could swap the last two bells:

        1 2 3 4
            ---
        1 2 4 3
        

        Each permutation of the four bells is called a row: we have just rung two rows. Now you want to keep going: but:

        • you don’t want to repeat any rows (permutations)
        • your bells are still limited to either swapping places with their neighbour, or staying in place.

        and to that you add these artistic constraints:

        • you want to ring every permutation of your 4 or 6 bells, or perhaps half or a quarter of all possible permutations, without repeating any permutation
        • you want your ringers to be able to remember what to ring – they’re not going to memorize the 6! = 720 permutations, and the place of their bell in each permutation. (A conductor who “calls the changes” helps here; but you’d still want large stretches to be predictable, so that the conductor can limit themself to calling the special bits.)

        So you need a predictable pattern (of, say, 24 rows) of which bells stay in place, and which bells swap; and when you reach the end, you swap two bells and repeat the pattern with a different starting row, so you get another 24 permutations out of it; and you do that again until you’ve finally rung all possible rows (or an eighth/quarter/half of them). That pattern is also called a method, and because it determines the sequence of notes you hear you could also call it a compostion.

        All the above is explained much better in the video @tgfrerer linked, which is shortish and enjoyable.

        Compositions have exciting names like “Kent Treble Bob” or “Double Oxford Minors” or “Grandsire Triples”.

        For fun, and because we have an infinite canvas here, here’s you could go through all 24 permutations of 4 bells by swapping exactly one pair each round. Another method/composition could have you swapping sometimes two bells, sometimes one; and with six bells there’s even more possible ways to generate subsequent permutations, eventually generating all of them.

         1 2 3 4   The first row of a peal is traditionally from high to low, a.k.a. rounds
             --- } Bell 4 hunts down (towards the front)
         1 2 4 3 }
           ---   }
         1 4 2 3 }
         ---     }
         4 1 2 3
             ---   The back end swaps
         4 1 3 2
         ---     } Bell 4 hunts up (towards the back)
         1 4 3 2 }
           ---   }
         1 3 4 2 }
             --- }
         1 3 2 4
         ---     } The front end swaps
         3 1 2 4
             --- } Bell 4 hunts down
         3 1 4 2 }
           ---   }
         3 4 1 2 }
         ---     }
         4 3 1 2
             ---   The back end swaps
         4 3 2 1
         ---     } Bell 4 hunts up
         3 4 2 1 }
           ---   }
         3 2 4 1 }
             --- }
         3 2 1 4
         ---     } The front end swaps
         2 3 1 4
             --- } Bell 4 hunts down
         2 3 4 1 }
           ---   }
         2 4 3 1 }
         ---     }
         4 2 3 1
             ---   The back end swaps
         4 2 1 3
         ---     } Bell 4 hunts up
         2 4 1 3 }
           ---   }
         2 1 4 3 }
             --- }
         2 1 3 4  And that's the last of the 24 possible permutations!
        (---    ) If we'd continue the pattern, we'd be back where we started
        (1 2 3 4)
        
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        This is good stuff, but there’s so much more… One of my favorite books, The Self-Made Tapestry by Phillip Ball, surveys an impressive range of basic processes. Most of them can be modeled in software, at least crudely.

        Why are snowflakes so symmetrical? Sure, basic hexagonal symmetry is due to the shape of the water molecule. But how does information get distributed between branches during growth? Last I checked, there were models that almost but couldn’t quite definitively answer that. There’s so much we still don’t know… and that’s before we start considering even the simplest biological processes.

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          Wow, what a fantastic reference, thank you very much! It would fit very nicely on the shelf next to On Growth and Form by D’Arcy Wentworth Thompson, I should think :)