Symmetry

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The Life transition rule, like that of any outer-totalistic cellular automaton, is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its neighbourhood 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.

Overview of symmetries.

Rotational symmetries

Rotational symmetries include the following (note that "C" refers to the cyclic groups):

C1

C1: Symmetric under 360° rotation. This is essentially no symmetry at all.

C1 symmetry

C2

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.
C2_1 symmetry
C2_2 symmetry
C2_4 symmetry

C4

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.
C4_1 symmetry
C4_4 symmetry

Reflectional symmetries

Reflectional symmetries include the following (note that "D" refers to the dihedral groups):

D2

D2: Symmetric under reflection through a line. There are two possibilities:

  • D2_+ The line is horizontal or vertical. There are two sub-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_+1 symmetry
D2_+2 symmetry
  • D2_x The line is diagonal.
D2_x symmetry

D4

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 sub-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_+1 symmetry
D4_+2 symmetry
D4_+4 symmetry
  • D4_x The lines are diagonal. There are two sub-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.
D4_x1 symmetry
D4_x4 symmetry

D8

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.
D8_1 symmetry
D8_4 symmetry

References