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   The different symmetries possible in change-ringing methods Snowflake
I became interested in method-symmetries when looking at cyclic methods, which cannot have conventional symmetry. Martin Bright has written the definitive article on method-symmetry, which can be found on his homepage.

As Martin's article explains, a method with an even length can have one of eleven possible symmetry types; however, only nine are possible for valid, true methods. The following is a list of the nine, with details and an example grid for each type.

  • Name: Conventional symmetry (vertical symmetry about a change / palindromic symmetry)
    This is by far the most common type of symmetry in methods.

  • Example given: Single Oxford Bob Major
  • Notation: -14-16-18-18-18-16-14-12
    The notation is palindromic (reads the same forwards or backwards) about the mid-point (the half-lead), with the exception of the leadhead change.

  • Other Examples: Plain Bob, Cambridge Surprise

  • Name: Glide symmetry (skew symmetry)
    First used in 1752, it has become popular recently due to its use in cyclic methods. Methods with this symmetry have no distinct reverse.

  • Example given: Double Cambridge Cyclic Bob
  • Notation: -16-16-18-78-38-38-18-12
    After the half-lead, the notation from the first half is used from the beginning in the same order, but reversed.

  • Other Examples: Double Eastern Bob, Olympus Surprise

  • Name: Rotational symmetry
    This has also been used recently in cyclic methods. The grid is the same when rotated through 180 degrees.

  • Example given: Anglia Cyclic Bob
  • Notation: -18-123678-18-78-58-36-14-12
    Moving out both ways from either quarter-lead (here the 123678 and the 36), the notations are the reverses of each other.

  • Other Examples: Brave New World Cyclic Bob, Nimrod Surprise.

  • Name: Horizontal symmetry
    A method with only this type of symmetry has never been rung. Methods must be twin-hunt with this symmetry.

  • Example given: Unnamed Bob Major
  • Notation: -18-36-1458-1458-18-18-18-1278
    Only horizontally symmetric notations are allowed. On eight bells, this means only x,18,1278,1458,123678 & 36 are allowed.

  • Other Examples: -18-36-18-18-18-18-18-1278

  • Name: Mirror symmetry
    This is horizontal symmetry, with conventional (palindromic) symmetry as well. Again. methods must be twin-hunt with this symmetry.

  • Example given: Bucknell Bob Major
  • Notation: -18-36-
    Again, only horizontally symmetric notations are allowed.

  • Other Examples: Mirror Bob

  • Name: Double symmetry
    This is conventional (palindromic) symmetry as well, with rotational symmetry about each quarter-lead.

  • Example given: Double Oxford Bob Major
  • Notation: -14-36-58-78-58-36-14-12
    The notation can be thought of as conventional symmetry with quarter-lead rotations.

  • Other Examples: Bristol Surprise, Double Bob

  • Name: Vertical (about a row)
    This is only one valid method possible with this symmetry.

  • Example given: Cross Differential
  • Notation: -
    The method is symmetrically very pure.

  • Other Examples: n/a

  • Name: Asymmetric
    There is no symmetry present in these methods.

  • Example given: Eastern Bob
  • Notation: -14-38-18-18-18-18-18-12

  • Other Examples: Selenium Surprise, Quasar