Static symmetry

A static symmetry refers to the combined rotational and reflectional symmetries of an arrangement of cells on a grid. Most isotropic rules preserve all such symmetries.

This page specifically only covers purely geometric symmetries. Other preserved attributes which are not necessarily geometrical symmetries, notably gutters, are compiled on the Minor static symmetries page. Symmetries which apply to periodic objects, such as still lifes, oscillators and spaceships, which also includes time symmetries, are listed on the Kinetic symmetry page.

Basic theory

The Life transition rule, like that of any isotropic cellular automaton, is invariant under valid 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.

On a square grid

Overview of symmetries (excluding D8_2).

The directed graph above shows how the various static symmetries are related to each other. Each downward arrow corresponds to the removal of either a rotational symmetry or a line of reflection.

For an overview of static symmetries in both Hickerson notation and Catagolue notation, see Table of equivalent static symmetries or the discussion in the kinetic symmetry article.

Rotational

In Catagolue notation, rotational symmetries are prefixed with "C", referring to the cyclic groups.

C1

C1: Symmetric under 360° rotation. This is essentially no symmetry at all. Example: Eater 1

x = 16, y = 16, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 160 NOGUI ]]
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. Example: Long snake
• C2_2: Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: Aircraft carrier
• C2_4: Rotation around a corner of a cell. The bounding rectangle is even by even. Example: Snake

x = 31, y = 31, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$31o $15bo14bo$15bo14bo$15bo9b2o3bo$15bo8b4o2bo$15bo8b4o2bo$15bo2b10o2bo$ 15bo14bo$15bo14bo$15bo3b2o4b2o3bo$15bo2bo2bo2bo2bo2bo$15bo2bo2bo2bo2bo 2bo$15bo3b2o4b2o3bo$15bo14bo$15bo14bo$15b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] C2_1 symmetry

x = 31, y = 32, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o $15b16o$15bo14bo$15bo14bo$15bo9b2o3bo$15bo8b4o2bo$15bo8b4o2bo$15bo2b 10o2bo$15bo14bo$15bo14bo$15bo3b2o4b2o3bo$15bo2bo2bo2bo2bo2bo$15bo2bo2b o2bo2bo2bo$15bo3b2o4b2o3bo$15bo14bo$15bo14bo$15b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] C2_2 symmetry

x = 32, y = 32, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o $16b16o$16bo14bo$16bo14bo$16bo9b2o3bo$16bo8b4o2bo$16bo8b4o2bo$16bo2b 10o2bo$16bo14bo$16bo14bo$16bo3b2o4b2o3bo$16bo2bo2bo2bo2bo2bo$16bo2bo2b o2bo2bo2bo$16bo3b2o4b2o3bo$16bo14bo$16bo14bo$16b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] 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. Example: Shuriken
• C4_4: Rotation around a corner of a cell. The bounding rectangle is even by even. Example: Quad

x = 31, y = 31, rule = B/S01234V 31o$o14bo14bo$o14bo14bo$o3b2o4b2o3bo3b3o3b2o3bo$o2bo2bo2bo2bo2bo2b4o2b o2bo2bo$o2bo2bo2bo2bo2bo2b4o2bo2bo2bo$o3b2o4b2o3bo3b3o3b2o3bo$o14bo5bo 8bo$o14bo5bo8bo$o2b10o2bo5bo3b2o3bo$o2b4o8bo5bo2bo2bo2bo$o2b4o8bo5bo2b o2bo2bo$o3b2o9bo5bo3b2o3bo$o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo $o3b2o3bo5bo9b2o3bo$o2bo2bo2bo5bo8b4o2bo$o2bo2bo2bo5bo8b4o2bo$o3b2o3bo 5bo2b10o2bo$o8bo5bo14bo$o8bo5bo14bo$o3b2o3b3o3bo3b2o4b2o3bo$o2bo2bo2b 4o2bo2bo2bo2bo2bo2bo$o2bo2bo2b4o2bo2bo2bo2bo2bo2bo$o3b2o3b3o3bo3b2o4b 2o3bo$o14bo14bo$o14bo14bo$31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] C4_1 symmetry

x = 32, y = 32, rule = B/S01234V 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b3o3b2o3bo$o2bo2bo2bo2bo2b2o2b 4o2bo2bo2bo$o2bo2bo2bo2bo2b2o2b4o2bo2bo2bo$o3b2o4b2o3b2o3b3o3b2o3bo$o 14b2o5bo8bo$o14b2o5bo8bo$o2b10o2b2o5bo3b2o3bo$o2b4o8b2o5bo2bo2bo2bo$o 2b4o8b2o5bo2bo2bo2bo$o3b2o9b2o5bo3b2o3bo$o14b2o14bo$o14b2o14bo$32o$32o $o14b2o14bo$o14b2o14bo$o3b2o3bo5b2o9b2o3bo$o2bo2bo2bo5b2o8b4o2bo$o2bo 2bo2bo5b2o8b4o2bo$o3b2o3bo5b2o2b10o2bo$o8bo5b2o14bo$o8bo5b2o14bo$o3b2o 3b3o3b2o3b2o4b2o3bo$o2bo2bo2b4o2b2o2bo2bo2bo2bo2bo$o2bo2bo2b4o2b2o2bo 2bo2bo2bo2bo$o3b2o3b3o3b2o3b2o4b2o3bo$o14b2o14bo$o14b2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] C4_4 symmetry

C4 tends to produce record diehards and megasized soups compared to other symmetries.

Reflectional

In Catagolue notation, reflectional symmetries are prefixed with "D", referring 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. Example: Hat
  • D2_+2 The line lies between two rows of cells. The bounding rectangle is even by any. Example: Frutterfly

x = 16, y = 31, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o $o14bo$o14bo$o3b2o9bo$o2b4o8bo$o2b4o8bo$o2b10o2bo$o14bo$o14bo$o3b2o4b 2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b2o3bo$o14bo$o14bo$16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 320 NOGUI ]] D2_+1 symmetry

x = 16, y = 32, rule = B/S01234V 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o $16o$o14bo$o14bo$o3b2o9bo$o2b4o8bo$o2b4o8bo$o2b10o2bo$o14bo$o14bo$o3b 2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b2o3bo$o14bo$o14bo$ 16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 320 NOGUI ]] D2_+2 symmetry

• D2_x The line is diagonal. Example: Elevener

x = 16, y = 16, rule = B/S01234V 16o$o14bo$o14bo$o7bo6bo$o6bobo5bo$o7bo3bo2bo$o11bo2bo$o3bo7bo2bo$o2bob o6bo2bo$o3bo6bo3bo$o9bo4bo$o8bo5bo$o4b4o6bo$o14bo$o14bo$16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 160 NOGUI ]]
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. Example: Dead spark coil
  • D4_+2: Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: Honeycomb
  • D4_+4: Rotation around a corner of a cell. The bounding rectangle is even by even. Example: A for all

x = 31, y = 31, rule = B/S01234V 31o$o14bo14bo$o14bo14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo 2bo2bo2bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14b o14bo$o14bo14bo$o2b10o2bo2b10o2bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o3b 2o9bo9b2o3bo$o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo$o3b2o9bo9b2o 3bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o2b10o2bo2b10o2bo$o14bo14bo$o14bo 14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o2bo2bo2b o2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14bo14bo$o14bo14bo$31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+1 symmetry

x = 32, y = 31, rule = B/S01234V 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o 8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o$o14b2o14bo$o14b2o 14bo$o3b2o9b2o9b2o3bo$o2b4o8b2o8b4o2bo$o2b4o8b2o8b4o2bo$o2b10o2b2o2b 10o2bo$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b 2o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b 2o3bo$o14b2o14bo$o14b2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 symmetry

x = 32, y = 32, rule = B/S01234V 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o 8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b 2o14bo$o3b2o9b2o9b2o3bo$o2b4o8b2o8b4o2bo$o2b4o8b2o8b4o2bo$o2b10o2b2o2b 10o2bo$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b 2o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b 2o3bo$o14b2o14bo$o14b2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] 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. Example: Loaf siamese loaf
  • D4_x4: Rotation around a corner of a cell. The bounding rectangle is even by even. Example: Long ship

x = 31, y = 31, rule = B/S01234V 31o$o14bo14bo$o14bo14bo$o7bo6bo5b3o6bo$o6bobo5bo5bobo6bo$o7bo3bo2bo2bo 2b3o6bo$o11bo2bo2b2o10bo$o3bo7bo2bo2b2o5b3o2bo$o2bobo6bo2bo2b3o4bobo2b o$o3bo6bo3bo3b2o4b3o2bo$o9bo4bo3b4o7bo$o8bo5bo4b5o5bo$o4b4o6bo6b4o4bo$ o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo$o4b4o6bo6b4o4bo$o5b5o4bo5b o8bo$o7b4o3bo4bo9bo$o2b3o4b2o3bo3bo6bo3bo$o2bobo4b3o2bo2bo6bobo2bo$o2b 3o5b2o2bo2bo7bo3bo$o10b2o2bo2bo11bo$o6b3o2bo2bo2bo3bo7bo$o6bobo5bo5bob o6bo$o6b3o5bo6bo7bo$o14bo14bo$o14bo14bo$31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_x1 symmetry

x = 32, y = 32, rule = B/S01234V 32o$o14b2o14bo$o14b2o14bo$o7bo6b2o5b3o6bo$o6bobo5b2o5bobo6bo$o7bo3bo2b 2o2bo2b3o6bo$o11bo2b2o2b2o10bo$o3bo7bo2b2o2b2o5b3o2bo$o2bobo6bo2b2o2b 3o4bobo2bo$o3bo6bo3b2o3b2o4b3o2bo$o9bo4b2o3b4o7bo$o8bo5b2o4b5o5bo$o4b 4o6b2o6b4o4bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b2o14bo$o4b 4o6b2o6b4o4bo$o5b5o4b2o5bo8bo$o7b4o3b2o4bo9bo$o2b3o4b2o3b2o3bo6bo3bo$o 2bobo4b3o2b2o2bo6bobo2bo$o2b3o5b2o2b2o2bo7bo3bo$o10b2o2b2o2bo11bo$o6b 3o2bo2b2o2bo3bo7bo$o6bobo5b2o5bobo6bo$o6b3o5b2o6bo7bo$o14b2o14bo$o14b 2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] 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. Example: Pulsar
• D8_4: Rotation around a corner of a cell. The bounding rectangle is even by even. Example: Lake 2

x = 31, y = 31, rule = B/S01234V 31o$o14bo14bo$o14bo14bo$o7bo6bo6bo7bo$o6bobo5bo5bobo6bo$o7bo3bo2bo2bo 3bo7bo$o11bo2bo2bo11bo$o3bo7bo2bo2bo7bo3bo$o2bobo6bo2bo2bo6bobo2bo$o3b o6bo3bo3bo6bo3bo$o9bo4bo4bo9bo$o8bo5bo5bo8bo$o4b4o6bo6b4o4bo$o14bo14bo $o14bo14bo$31o$o14bo14bo$o14bo14bo$o4b4o6bo6b4o4bo$o8bo5bo5bo8bo$o9bo 4bo4bo9bo$o3bo6bo3bo3bo6bo3bo$o2bobo6bo2bo2bo6bobo2bo$o3bo7bo2bo2bo7bo 3bo$o11bo2bo2bo11bo$o7bo3bo2bo2bo3bo7bo$o6bobo5bo5bobo6bo$o7bo6bo6bo7b o$o14bo14bo$o14bo14bo$31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 symmetry

x = 32, y = 32, rule = B/S01234V 32o$o14b2o14bo$o14b2o14bo$o7bo6b2o6bo7bo$o6bobo5b2o5bobo6bo$o7bo3bo2b 2o2bo3bo7bo$o11bo2b2o2bo11bo$o3bo7bo2b2o2bo7bo3bo$o2bobo6bo2b2o2bo6bob o2bo$o3bo6bo3b2o3bo6bo3bo$o9bo4b2o4bo9bo$o8bo5b2o5bo8bo$o4b4o6b2o6b4o 4bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b2o14bo$o4b4o6b2o6b4o 4bo$o8bo5b2o5bo8bo$o9bo4b2o4bo9bo$o3bo6bo3b2o3bo6bo3bo$o2bobo6bo2b2o2b o6bobo2bo$o3bo7bo2b2o2bo7bo3bo$o11bo2b2o2bo11bo$o7bo3bo2b2o2bo3bo7bo$o 6bobo5b2o5bobo6bo$o7bo6b2o6bo7bo$o14b2o14bo$o14b2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_4 symmetry

On a hexagonal or triangular grid

Overview of hexagonal symmetries. Blue arrows correspond to non-normal subgroups.

Hexagonal and triangular grids have the same set of admissible symmetries as each other (by planar or polytopic duality - see also Schläfli symbol), but these are not the same symmetries as square grids. Due to how hexagonal and triangular grids are handled by programs such as Golly and LifeViewer, they will also appear markedly different in these respects. C2, D2, and D4 symmetries are still compatible, but C4 symmetries become meaningless because the cells no longer have a side count that is perfectly divisible by 4. Other symmetries are exclusive to these alternative grids, as indicated below:

• C1
• C2_1
• C2_2
• C3_1
• C3_3 (unsupported by apgsearch)
• C6
• D2_xo
• D2_x
• D4_x1
• D4_x4
• D6_1
• D6_1o
• D6_3 (unsupported by apgsearch)
• D12

apgsearch currently supports most higher symmetries for hexagonal rules; the rest (C3_3 and D6_3) will be added in a future version.

Rotational

Click on "Expand" to the right to view a list of hexagonal/triangular rotational symmetries.

C1

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

x = 16, y = 31, rule = B/S0123HT o$2o$obo$o2bo$o3bo$o4bo$o5bo$o6bo$o7bo$o8bo$o9bo$o2b2o6bo$o2bobo2bo3b o$o3b2o2bo4bo$o8bo4bo$o8bo5bo$bo8b2o3bo$2bo3b2o2b3o2bo$3bo2bobo2b2o2b o$4bo2b2o6bo$5bo9bo$6bo8bo$7bo7bo$8bo6bo$9bo5bo$10bo4bo$11bo3bo$12bo2b o$13bobo$14b2o$15bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 300 HEIGHT 300 NOGUI ]]
C1 symmetry

C2

C2: Symmetric under 180° rotation. There are two possibilities:

• C2_1: Rotation around the center of a cell.
• C2_4: Rotation around the midpoint of a side of a cell.

x = 31, y = 31, rule = B/S0123HT o14bo$2o13b2o$obo12bobo$o2bo11bo2bo$o3bo10bo3bo$o4bo9bo4bo$o5bo8bo5bo $o6bo7bo6bo$o7bo6bo7bo$o8bo5bo8bo$o9bo4bo9bo$o2b2o6bo3bo6b2o2bo$o2bob o2bo3bo2bo2b2o2bobo2bo$o3b2o2bo4bobo2b3o2b2o3bo$o8bo4b2o3b2o8bo$o8bo5b o5bo8bo$bo8b2o3b2o4bo8bo$2bo3b2o2b3o2bobo4bo2b2o3bo$3bo2bobo2b2o2bo2b o3bo2bobo2bo$4bo2b2o6bo3bo6b2o2bo$5bo9bo4bo9bo$6bo8bo5bo8bo$7bo7bo6bo 7bo$8bo6bo7bo6bo$9bo5bo8bo5bo$10bo4bo9bo4bo$11bo3bo10bo3bo$12bo2bo11b o2bo$13bobo12bobo$14b2o13b2o$15bo14bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] C2_1 symmetry

x = 32, y = 31, rule = B/S0123HT o15bo$2o14b2o$obo13bobo$o2bo12bo2bo$o3bo11bo3bo$o4bo10bo4bo$o5bo9bo5b o$o6bo8bo6bo$o7bo7bo7bo$o8bo6bo8bo$o9bo5bo9bo$o2b2o6bo4bo6b2o2bo$o2bo bo2bo3bo3bo2b2o2bobo2bo$o3b2o2bo4bo2bo2b3o2b2o3bo$o8bo4bobo3b2o8bo$o8b o5b2o5bo8bo$bo8b2o3bobo4bo8bo$2bo3b2o2b3o2bo2bo4bo2b2o3bo$3bo2bobo2b2o 2bo3bo3bo2bobo2bo$4bo2b2o6bo4bo6b2o2bo$5bo9bo5bo9bo$6bo8bo6bo8bo$7bo7b o7bo7bo$8bo6bo8bo6bo$9bo5bo9bo5bo$10bo4bo10bo4bo$11bo3bo11bo3bo$12bo2b o12bo2bo$13bobo13bobo$14b2o14b2o$15bo15bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] C2_4 symmetry

C3

C3: Symmetric under 120° rotation. There are two possibilities:

• C3_1: Rotation around the center of a cell.
• C3_3: Rotation around a corner of a cell. (unsupported by apgsearch)

x = 46, y = 46, rule = B/S0123HT o$2o$obo$o2bo$o3bo$o4bo$o5bo$o6bo$o7bo$o8bo$o9bo$o6b2o2bo$o2b2o2bobo2b o$o2b3o2b2o3bo$o3b2o8bo$o5bo8bo$bo4bo8bo$2bo4bo2b2o3bo$3bo3bo2bobo2bo$ 4bo6b2o2bo$5bo9bo$6bo8bo$7bo7bo$8bo6bo$9bo5bo$10bo4bo$11bo3bo$12bo2bo$ 13bobo$14b2o$31o$o14b2o14bo$o14bobo14bo$o5b2o7bo2bo14bo$o5bobo6bo3bo7b o6bo$o6b2o6bo4bo7b2ob2o2bo$o2b2o10bo5bo8b4o2bo$o2bobo9bo6bo3b2o4b2o3bo $o3b2o4b2o3bo7bo2bobo9bo$o8b2o4bo8bo2b2o10bo$o7b2o5bo9bo6b2o6bo$o7b3o 4bo10bo5bobo6bo$o8b2o4bo11bo5b2o7bo$o14bo12bo14bo$o14bo13bo14bo$16o14b 16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] C3_1 symmetry

x = 47, y = 47, rule = B/S0123HT o$2o$obo$o2bo$o3bo$o4bo$o5bo$o6bo$o7bo$o8bo$o9bo$o6b2o2bo$o2b2o2bobo2b o$o2b3o2b2o3bo$o3b2o8bo$o5bo8bo$bo4bo8bo$2bo4bo2b2o3bo$3bo3bo2bobo2bo $4bo6b2o2bo$5bo9bo$6bo8bo$7bo7bo$8bo6bo$9bo5bo$10bo4bo$11bo3bo$12bo2b o$13bobo$14b2o$15bo$32o$o14bobo14bo$o14bo2bo14bo$o5b2o7bo3bo14bo$o5bo bo6bo4bo7bo6bo$o6b2o6bo5bo7b2ob2o2bo$o2b2o10bo6bo8b4o2bo$o2bobo9bo7bo 3b2o4b2o3bo$o3b2o4b2o3bo8bo2bobo9bo$o8b2o4bo9bo2b2o10bo$o7b2o5bo10bo6b 2o6bo$o7b3o4bo11bo5bobo6bo$o8b2o4bo12bo5b2o7bo$o14bo13bo14bo$o14bo14b o14bo$16o15b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] C3_3 symmetry

C6

C6: Symmetric under 60° rotation.

x = 61, y = 61, rule = B/S0123HT 15bo$15b2o$15bobo$15bo2bo$15bo3bo$15bo4bo$15bo5bo$15bo6bo$15bo7bo$15bo 8bo$15bo9bo$15bo6b2o2bo$15bo2b2o2bobo2bo$15bo2b3o2b2o3bo$15bo3b2o8bo$ 16o5bo8b16o$bo14bo4bo8bo14bo$2bo14bo4bo2b2o3bo14bo$3bo7b2o5bo3bo2bobo 2bo4b2o8bo$4bo6bobo5bo6b2o2bo4b3o7bo$5bo6b2o6bo9bo5b2o7bo$6bo10b2o2bo 8bo4b2o8bo$7bo9bobo2bo7bo3b2o4b2o3bo$8bo3b2o4b2o3bo6bo9bobo2bo$9bo2b4o 8bo5bo10b2o2bo$10bo2b2ob2o7bo4bo6b2o6bo$11bo6bo7bo3bo6bobo5bo$12bo14bo 2bo7b2o5bo$13bo14bobo14bo$14bo14b2o14bo$15b31o$15bo14b2o14bo$15bo14bob o14bo$15bo5b2o7bo2bo14bo$15bo5bobo6bo3bo7bo6bo$15bo6b2o6bo4bo7b2ob2o2b o$15bo2b2o10bo5bo8b4o2bo$15bo2bobo9bo6bo3b2o4b2o3bo$15bo3b2o4b2o3bo7bo 2bobo9bo$15bo8b2o4bo8bo2b2o10bo$15bo7b2o5bo9bo6b2o6bo$15bo7b3o4bo2b2o 6bo5bobo6bo$15bo8b2o4bo2bobo2bo3bo5b2o7bo$15bo14bo3b2o2bo4bo14bo$15bo 14bo8bo4bo14bo$15b16o8bo5b16o$31bo8b2o3bo$32bo3b2o2b3o2bo$33bo2bobo2b 2o2bo$34bo2b2o6bo$35bo9bo$36bo8bo$37bo7bo$38bo6bo$39bo5bo$40bo4bo$41bo 3bo$42bo2bo$43bobo$44b2o$45bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 500 HEIGHT 500 NOGUI ]]

C6 symmetry

Reflectional

Click on "Expand" to the right to view a list of hexagonal/triangular reflectional symmetries.

D2

D2: There is line symmetry. There are two possibilities:

• D2_x: Through the vertices of a cell (diagonal).
• D2_xo: Through the edges of a cell (orthogonal).

x = 16, y = 31, rule = B/S0123HT o$2o$obo$o2bo$o3bo$o4bo$o5bo$o6bo$o7bo$o8bo$o9bo$o2b2o2b2o2bo$o2bobob obo2bo$o3b2o2b2o3bo$o13bo$o14bo$bo13bo$2bo2bo6bo2bo$3bo2bo5bo2bo$4bo2b o4bo2bo$5bo2b5o2bo$6bo8bo$7bo7bo$8bo6bo$9bo5bo$10bo4bo$11bo3bo$12bo2b o$13bobo$14b2o$15bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 300 HEIGHT 300 NOGUI ]] D2_x symmetry

x = 16, y = 31, rule = B/S0123HT o$2o$obo$o2bo$o3bo$o4bo$o5bo$o6bo$o4bo2bo$o5bo2bo$o2b2o2bo2bo$o2bobo2b o2bo$o3b2o3bo2bo$o8bo3bo$o9bo3bo$o9bo4bo$bo9bo3bo$2bo8bo3bo$3bo3b2o3b o2bo$4bo2bobo2bo2bo$5bo2b2o2bo2bo$6bo5bo2bo$7bo4bo2bo$8bo6bo$9bo5bo$10b o4bo$11bo3bo$12bo2bo$13bobo$14b2o$15bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 300 HEIGHT 300 NOGUI ]] D2_xo 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_x1: Rotation around the center of a cell.
• D4_x4: Rotation around the edges of a cell.

x = 31, y = 31, rule = B/S0123HT o14bo$2o13b2o$obo12bobo$o2bo11bo2bo$o3bo10bo3bo$o4bo9bo4bo$o5bo8bo5bo $o6bo7bo6bo$o4bo2bo6bo2bo4bo$o5bo2bo5bo2bo5bo$o2b2o2bo2bo4bo2bo2b2o2b o$o2bobo2bo2bo3bo2bo2bobo2bo$o3b2o3bo2bo2bo2bo3b2o3bo$o8bo3bobo3bo8bo $o9bo3b2o3bo9bo$o9bo4bo4bo9bo$bo9bo3b2o3bo9bo$2bo8bo3bobo3bo8bo$3bo3b 2o3bo2bo2bo2bo3b2o3bo$4bo2bobo2bo2bo3bo2bo2bobo2bo$5bo2b2o2bo2bo4bo2b o2b2o2bo$6bo5bo2bo5bo2bo5bo$7bo4bo2bo6bo2bo4bo$8bo6bo7bo6bo$9bo5bo8bo 5bo$10bo4bo9bo4bo$11bo3bo10bo3bo$12bo2bo11bo2bo$13bobo12bobo$14b2o13b 2o$15bo14bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] D4_x1 symmetry

x = 32, y = 31, rule = B/S0123HT o15bo$2o14b2o$obo13bobo$o2bo12bo2bo$o3bo11bo3bo$o4bo10bo4bo$o5bo9bo5b o$o6bo8bo6bo$o4bo2bo7bo2bo4bo$o5bo2bo6bo2bo5bo$o2b2o2bo2bo5bo2bo2b2o2b o$o2bobo2bo2bo4bo2bo2bobo2bo$o3b2o3bo2bo3bo2bo3b2o3bo$o8bo3bo2bo3bo8b o$o9bo3bobo3bo9bo$o9bo4b2o4bo9bo$bo9bo3bobo3bo9bo$2bo8bo3bo2bo3bo8bo$ 3bo3b2o3bo2bo3bo2bo3b2o3bo$4bo2bobo2bo2bo4bo2bo2bobo2bo$5bo2b2o2bo2bo 5bo2bo2b2o2bo$6bo5bo2bo6bo2bo5bo$7bo4bo2bo7bo2bo4bo$8bo6bo8bo6bo$9bo5b o9bo5bo$10bo4bo10bo4bo$11bo3bo11bo3bo$12bo2bo12bo2bo$13bobo13bobo$14b 2o14b2o$15bo15bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] D4_x4 symmetry

D6

D6: Symmetric under both reflection and 120° rotation. The reflection symmetry will be with respect to three lines. There are three possibilities:

• D6_1: Rotation around the center of a cell with lines going through the vertices of cells.
• D6_1o: Rotation around the center of a cell with lines going through the edges of cells.
• D6_3: Rotation around the corner of a cell. (unsupported by apgsearch)

x = 46, y = 46, rule = B/S0123HT 16o14b16o$bo14bo13bo14bo$2bo14bo12bo14bo$3bo6b4o4bo11bo4b4o6bo$4bo5bo 8bo10bo8bo5bo$5bo4bo9bo9bo9bo4bo$6bo3bo6b2o2bo8bo2b2o6bo3bo$7bo2bo6bo bo2bo7bo2bobo6bo2bo$8bo2bo6b2o3bo6bo3b2o6bo2bo$9bo2bo11bo5bo11bo2bo$10b o2bo3b2o6bo4bo6b2o3bo2bo$11bo5bobo6bo3bo6bobo5bo$12bo5b2o7bo2bo7b2o5b o$13bo14bobo14bo$14bo14b2o14bo$15b31o$30b2o$30bobo$30bo2bo$30bo3bo$30b o4bo$30bo5bo$30bo6bo$30bo7bo$30bo8bo$30bo9bo$30bo2b2o2b2o2bo$30bo2bob obobo2bo$30bo3b2o2b2o3bo$30bo13bo$30bo14bo$31bo13bo$32bo2bo6bo2bo$33b o2bo5bo2bo$34bo2bo4bo2bo$35bo2b5o2bo$36bo8bo$37bo7bo$38bo6bo$39bo5bo$ 40bo4bo$41bo3bo$42bo2bo$43bobo$44b2o$45bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] D6_1 symmetry

x = 31, y = 31, rule = B/S0123HT 16o$2o14bo$obo14bo$o2bo4b5o5bo$o3bo8b2o4bo$o4bo9b2o3bo$o5bo3b2o5b2o2bo $o6bo2bobo6bo2bo$o2bo4bo2b2o7bo2bo$o2bo5bo11bo2bo$o2bo2b2o2bo7b2o2bo2b o$o2bo2bobo2bo6bobo5bo$o2bo3b2o3bo6b2o6bo$o3bo8bo14bo$o3bo9bo14bo$o4bo 9b16o$bo3bo9bo14bo$2bo3bo8bo14bo$3bo2bo3b2o3bo6b2o6bo$4bo2bo2bobo2bo6b obo5bo$5bo2bo2b2o2bo7b2o2bo2bo$6bo2bo5bo11bo2bo$7bo2bo4bo2b2o7bo2bo$8b o6bo2bobo6bo2bo$9bo5bo3b2o5b2o2bo$10bo4bo9b2o3bo$11bo3bo8b2o4bo$12bo2b o4b5o5bo$13bobo14bo$14b2o14bo$15b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] D6_1o symmetry

x = 47, y = 47, rule = B/S0123HT 16o15b16o$bo14bo14bo14bo$2bo14bo13bo14bo$3bo6b4o4bo12bo4b4o6bo$4bo5bo 8bo11bo8bo5bo$5bo4bo9bo10bo9bo4bo$6bo3bo6b2o2bo9bo2b2o6bo3bo$7bo2bo6b obo2bo8bo2bobo6bo2bo$8bo2bo6b2o3bo7bo3b2o6bo2bo$9bo2bo11bo6bo11bo2bo$ 10bo2bo3b2o6bo5bo6b2o3bo2bo$11bo5bobo6bo4bo6bobo5bo$12bo5b2o7bo3bo7b2o 5bo$13bo14bo2bo14bo$14bo14bobo14bo$15b32o$31bo$31b2o$31bobo$31bo2bo$31b o3bo$31bo4bo$31bo5bo$31bo6bo$31bo7bo$31bo8bo$31bo9bo$31bo2b2o2b2o2bo$ 31bo2bobobobo2bo$31bo3b2o2b2o3bo$31bo13bo$31bo14bo$32bo13bo$33bo2bo6b o2bo$34bo2bo5bo2bo$35bo2bo4bo2bo$36bo2b5o2bo$37bo8bo$38bo7bo$39bo6bo$ 40bo5bo$41bo4bo$42bo3bo$43bo2bo$44bobo$45b2o$46bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 400 HEIGHT 400 NOGUI ]] D6_3 symmetry

D12

D12: Symmetric under both reflection and 60° rotation. The reflection symmetry will be with respect to six lines.

x = 61, y = 61, rule = B/S0123HT 15bo$15b2o$15bobo$15bo2bo$15bo3bo$15bo4bo$15bo5bo$15bo6bo$15bo7bo$15bo 8bo$15bo2b5o2bo$15bo2bo4bo2bo$15bo2bo5bo2bo$15bo2bo6bo2bo$15bo13bo$16o 14b16o$bo14bo13bo14bo$2bo14bo3b2o2b2o3bo14bo$3bo6b4o4bo2bobobobo2bo4b 4o6bo$4bo5bo8bo2b2o2b2o2bo8bo5bo$5bo4bo9bo9bo9bo4bo$6bo3bo6b2o2bo8bo2b 2o6bo3bo$7bo2bo6bobo2bo7bo2bobo6bo2bo$8bo2bo6b2o3bo6bo3b2o6bo2bo$9bo2b o11bo5bo11bo2bo$10bo2bo3b2o6bo4bo6b2o3bo2bo$11bo5bobo6bo3bo6bobo5bo$ 12bo5b2o7bo2bo7b2o5bo$13bo14bobo14bo$14bo14b2o14bo$15b31o$15bo14b2o14b o$15bo14bobo14bo$15bo5b2o7bo2bo7b2o5bo$15bo5bobo6bo3bo6bobo5bo$15bo2bo 3b2o6bo4bo6b2o3bo2bo$15bo2bo11bo5bo11bo2bo$15bo2bo6b2o3bo6bo3b2o6bo2bo $15bo2bo6bobo2bo7bo2bobo6bo2bo$15bo3bo6b2o2bo8bo2b2o6bo3bo$15bo4bo9bo 9bo9bo4bo$15bo5bo8bo2b2o2b2o2bo8bo5bo$15bo6b4o4bo2bobobobo2bo4b4o6bo$ 15bo14bo3b2o2b2o3bo14bo$15bo14bo13bo14bo$15b16o14b16o$31bo13bo$32bo2bo 6bo2bo$33bo2bo5bo2bo$34bo2bo4bo2bo$35bo2b5o2bo$36bo8bo$37bo7bo$38bo6bo $39bo5bo$40bo4bo$41bo3bo$42bo2bo$43bobo$44b2o$45bo! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 500 HEIGHT 500 NOGUI ]]

D12 symmetry

Higher Dimensions

The symmetries for the cubic grid are listed at Cubic grid symmetries.

External links

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