Static symmetry

The Life transition rule, like that of any isotropic 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.

Square-grid symmetries

Overview of symmetries (excluding D8_2).

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 orthogonal. 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 orthogonal. 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_2: Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
• D8_4: Rotation around a corner of a cell. The bounding rectangle is even by even.

D8_1 symmetry

D8_2 symmetry

D8_4 symmetry

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should (incomplete, please verify and complete):

• B1c/B2c/B4c
• B1e/B2a/B2i/B4i
• B3e/B3j
• B3i/B6i
• B3q/B3y
• B4t/B5r
• B4w/B5y/B4e
• B5a/B7e/B8
• B6c/B6k
• S0/S1e/S3a
• S2c/S2k
• S2i/S5i
• S3i/S4a/S4t/S6i/S7e
• S3r/S4i

Skew symmetries

If a pattern exhibits symmetry only after its constituent congruent pieces are offset by certain amounts in one or both orthogonal directions, the pattern is said to exhibit skew symmetry.

Gutter symmetries

Gutter symmetries are distinguished from non-gutter symmetries by the existence of an empty lane of cells – the "gutter" – separating the congruent pieces making up overall pattern.

A pattern that exhibits gutter symmetry only after its pieces are skewed in the above sense is said to exhibit skew-gutter symmetry.

Gutter and skewgutter symmetries are known to exist for both orthogonal and diagonal lines of symmetry......

Orthogonal gutter symmetry

Orthogonal skewgutter symmetry

Orthogonal double skewgutter symmetry

Diagonal gutter symmetry

Diagonal skewgutter symmetry

In order to preserve orthogonal gutter symmetry, the birth conditions B0, B2c, B2i, B4i, B4c and B6i must be absent.

In order to preserve orthogonal skewgutter symmetry, the birth conditions B0, B1c, B2k, B2n, B3n, B3y, B4y, B4z, B5r and B6i must be absent.

In order to preserve orthogonal double skewgutter symmetry, the birth conditions B0, B1c, B1e, B2a, B2i, B2k, B2n, B3c, B3q, B3r, B4c, B4n, B4y, B4z, B5e, B5r and B6i must be absent.

In order to preserve diagonal gutter symmetry, the birth conditions B0, B2n, B2e, B4e, B4w and B6n must be absent.

In order to preserve diagonal skewgutter symmetry, the birth consitions B0, B1c, B1e, B2a, B2k, B3k, B3q and B4q must be absent.

Hexagonal-grid and triangular-grid symmetries

Hexagonal and triangular grids do not have the same symmetries as square grids. C2, D2, and D4 symmetries are still compatible, but C4 symmetries become meaningless because the cells no longer have a side count that is divisible by 4. Other symmetries are exclusive to these alternative grids, such as C3_1, C3_3, D6_1, and D6_3. C6 and D12 symmetries are also possible in a hexagon-tiled universe.

apgsearch does not currently support all higher symmetries for hexagonal rules, but all will be added in a future version.

External links

• Help with symmetries (discussion thread) at the ConwayLife.com forums