Symmetry

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The Life transition rule, like that of any totalistic cellular automaton, is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its neighborhood is rotated or reflected. This implies there are symmetries which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

Rotation symmetries include the following:

  • C1: Symmetric under 360° rotation. This is essentially no symmetry at all.
  • C2: Symmetric under 180° rotation. There are three possibilities:
    • C2_1: Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd.
    • C2_2: Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
    • C2_4: Rotation around a corner of a cell. The bounding rectangle is even by even.
  • C4: Symmetric under 90° rotation. There are two possibilities:
    • C4_1: Rotation around the center of a cell. The bounding rectangle is odd by odd.
    • C4_4: Rotation around a corner of a cell. The bounding rectangle is even by even.

("C" refers to the cyclic group.)

Reflection symmetries include:

  • D2: Symmetric under reflection through a line. There are two possibilities:
    • D2_+ The line is horizontal or vertical. There are two possibilities:
      • D2_+1 The line bisects a row of cells. The bounding rectangle is odd by any.
      • D2_+2 The line lies between two rows of cells. The bounding rectangle is even by any.
    • D2_x The line is diagonal.
  • D4: Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:
    • D4_+: The lines are horizontal and vertical. There are three possibilities:
      • D4_+1: Rotation around the center of a cell. The bounding rectangle is odd by odd.
      • D4_+2: Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
      • D4_+4: Rotation around a corner of a cell. The bounding rectangle is even by even.
    • D4_x The lines are diagonal. There are two possibilities:
      • D4_x1: Rotation around the center of a cell. The bounding rectangle is odd by odd.
      • D4_x4: Rotation around a corner of a cell. The bounding rectangle is even by even.
  • D8: Symmetric under both reflection and 90° rotation. The reflection symmetry will be with respect to horizontal, vertical, and diagonal lines. There are two possibilities:
    • D8_1: Rotation around the center of a cell. The bounding rectangle is odd by odd.
    • D8_4: Rotation around a corner of a cell. The bounding rectangle is even by even.

("D" refers to the dihedral group.)

References