Symmetry
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.
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.
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.
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.
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_x The line is diagonal.
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_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.
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.
References
- Help with symmetries (discussion thread) at the ConwayLife.com forums