Difference between revisions of "Symmetry"
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Revision as of 15:44, 18 April 2016
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.
 D2_+ The line is horizontal or vertical. There are two possibilities:
 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.
 D4+: The lines are horizontal and vertical. There are three possibilities:
 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.)