Minor static symmetry

This page documents minor static symmetries, a subclass of static symmetries which concern significant attributes conserved by any qualifying pattern in a given rule, or group thereof, which are not rotational or reflectional geometric symmetries.

Skew symmetry

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

It is not currently known if any rules support patterns with non-trivial skew symmetry, outside of trivial cases. However, skew symmetry can be combined with gutter symmetry (see section below) to form skew-gutter symmetry.

Gutter symmetry

Gutter symmetries are distinguished from non-gutter symmetries by the existence of an empty lane of cells – referred to as a 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.

Trivially, 0-dimensional gutters can also be considered for sufficiently symmetric patterns, but these are not interesting.

On a square grid

Gutter and skewgutter symmetries are known to exist for both orthogonal and diagonal lines of symmetry. Click on "Expand" to the right to view a list of them.

It is not known if this list is exhaustive.

x = 16, y = 33, rule = B/S012345678 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o 2$16o$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 ]] Orthogonal gutter symmetry D2_+1_gO1S0

x = 17, y = 33, rule = B/S012345678 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o 2$b16o$bo14bo$bo14bo$bo3b2o9bo$bo2b4o8bo$bo2b4o8bo$bo2b10o2bo$bo14bo$b o14bo$bo3b2o4b2o3bo$bo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2bo$bo3b2o4b2o3bo$ bo14bo$bo14bo$b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 320 NOGUI ]] Orthogonal skewgutter symmetry D2_+1_gO1S1

x = 18, y = 33, rule = B/S012345678 16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b 2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o 2$2b16o$2bo14bo$2bo14bo$2bo3b2o9bo$2bo2b4o8bo$2bo2b4o8bo$2bo2b10o2bo$ 2bo14bo$2bo14bo$2bo3b2o4b2o3bo$2bo2bo2bo2bo2bo2bo$2bo2bo2bo2bo2bo2bo$ 2bo3b2o4b2o3bo$2bo14bo$2bo14bo$2b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 320 NOGUI ]] Orthogonal double skewgutter symmetry D2_+1_gO1S2

x = 31, y = 33, rule = B/S012345678 31o$o14bo14bo$o14bo14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo 2bo2bo2bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14b o14bo$o14bo14bo$o2b10o2bo2b10o2bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o3b 2o9bo9b2o3bo$o14bo14bo$o14bo14bo$31o2$31o$o14bo14bo$o14bo14bo$o3b2o9bo 9b2o3bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o2b10o2bo2b10o2bo$o14bo14bo$o 14bo14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o2bo 2bo2bo2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14bo14bo$o14bo14b o$31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0

x = 33, y = 33, rule = B/S012345678 16ob16o$o14bobo14bo$o14bobo14bo$o3b2o4b2o3bobo3b2o4b2o3bo$o2bo2bo2bo2b o2bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$o3b2o4b2o3bobo 3b2o4b2o3bo$o14bobo14bo$o14bobo14bo$o2b10o2bobo2b10o2bo$o2b4o8bobo8b4o 2bo$o2b4o8bobo8b4o2bo$o3b2o9bobo9b2o3bo$o14bobo14bo$o14bobo14bo$16ob 16o2$16ob16o$o14bobo14bo$o14bobo14bo$o3b2o9bobo9b2o3bo$o2b4o8bobo8b4o 2bo$o2b4o8bobo8b4o2bo$o2b10o2bobo2b10o2bo$o14bobo14bo$o14bobo14bo$o3b 2o4b2o3bobo3b2o4b2o3bo$o2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo 2bobo2bo2bo2bo2bo2bo$o3b2o4b2o3bobo3b2o4b2o3bo$o14bobo14bo$o14bobo14bo $16ob16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0

x = 32, y = 33, rule = B/S012345678 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o 8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o2$32o$o14b2o14bo$o 14b2o14bo$o3b2o9b2o9b2o3bo$o2b4o8b2o8b4o2bo$o2b4o8b2o8b4o2bo$o2b10o2b 2o2b10o2bo$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2b o2b2o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b 2o4b2o3bo$o14b2o14bo$o14b2o14bo$32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0

x = 33, y = 33, rule = B/S012345678 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o 8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o2$b32o$bo14b2o14bo$b o14b2o14bo$bo3b2o9b2o9b2o3bo$bo2b4o8b2o8b4o2bo$bo2b4o8b2o8b4o2bo$bo2b 10o2b2o2b10o2bo$bo14b2o14bo$bo14b2o14bo$bo3b2o4b2o3b2o3b2o4b2o3bo$bo2b o2bo2bo2bo2b2o2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$bo3b2o 4b2o3b2o3b2o4b2o3bo$bo14b2o14bo$bo14b2o14bo$b32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 with orthogonal gutter symmetry, skewed

x = 34, y = 33, rule = B/S012345678 32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o 8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o2$2b32o$2bo14b2o14bo $2bo14b2o14bo$2bo3b2o9b2o9b2o3bo$2bo2b4o8b2o8b4o2bo$2bo2b4o8b2o8b4o2bo $2bo2b10o2b2o2b10o2bo$2bo14b2o14bo$2bo14b2o14bo$2bo3b2o4b2o3b2o3b2o4b 2o3bo$2bo2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$2bo2bo2bo2bo2bo2b2o2bo2bo2bo 2bo2bo$2bo3b2o4b2o3b2o3b2o4b2o3bo$2bo14b2o14bo$2bo14b2o14bo$2b32o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 with orthogonal gutter symmetry, doubly skewed

x = 34, y = 34, rule = B/S012345678 17b16o$16obo14bo$o14bobo14bo$o14bobo3b2o4b2o3bo$o3b2o4b2o3bobo2bo2bo2b o2bo2bo$o2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bobo3b2o4b2o3b o$o3b2o4b2o3bobo14bo$o14bobo14bo$o14bobo2b10o2bo$o2b10o2bobo8b4o2bo$o 2b4o8bobo8b4o2bo$o2b4o8bobo9b2o3bo$o3b2o9bobo14bo$o14bobo14bo$o14bob 16o$16o$18b16o$b16obo14bo$bo14bobo14bo$bo14bobo9b2o3bo$bo3b2o9bobo8b4o 2bo$bo2b4o8bobo8b4o2bo$bo2b4o8bobo2b10o2bo$bo2b10o2bobo14bo$bo14bobo 14bo$bo14bobo3b2o4b2o3bo$bo3b2o4b2o3bobo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo 2bobo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2bobo3b2o4b2o3bo$bo3b2o4b2o3bobo14b o$bo14bobo14bo$bo14bob16o$b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+1 with two skew orthogonal gutters

x = 34, y = 33, rule = B/S012345678 16ob16o$o14bobo14bo$o14bobo14bo$o3b2o4b2o3bobo3b2o4b2o3bo$o2bo2bo2bo2b o2bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$o3b2o4b2o3bobo 3b2o4b2o3bo$o14bobo14bo$o14bobo14bo$o2b10o2bobo2b10o2bo$o2b4o8bobo8b4o 2bo$o2b4o8bobo8b4o2bo$o3b2o9bobo9b2o3bo$o14bobo14bo$o14bobo14bo$16ob 16o2$b16ob16o$bo14bobo14bo$bo14bobo14bo$bo3b2o9bobo9b2o3bo$bo2b4o8bobo 8b4o2bo$bo2b4o8bobo8b4o2bo$bo2b10o2bobo2b10o2bo$bo14bobo14bo$bo14bobo 14bo$bo3b2o4b2o3bobo3b2o4b2o3bo$bo2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$bo 2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$bo3b2o4b2o3bobo3b2o4b2o3bo$bo14bobo 14bo$bo14bobo14bo$b16ob16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 with two orthogonal gutters, one skewed

x = 35, y = 33, rule = B/S012345678 16ob16o$o14bobo14bo$o14bobo14bo$o3b2o4b2o3bobo3b2o4b2o3bo$o2bo2bo2bo2b o2bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$o3b2o4b2o3bobo 3b2o4b2o3bo$o14bobo14bo$o14bobo14bo$o2b10o2bobo2b10o2bo$o2b4o8bobo8b4o 2bo$o2b4o8bobo8b4o2bo$o3b2o9bobo9b2o3bo$o14bobo14bo$o14bobo14bo$16ob 16o2$2b16ob16o$2bo14bobo14bo$2bo14bobo14bo$2bo3b2o9bobo9b2o3bo$2bo2b4o 8bobo8b4o2bo$2bo2b4o8bobo8b4o2bo$2bo2b10o2bobo2b10o2bo$2bo14bobo14bo$ 2bo14bobo14bo$2bo3b2o4b2o3bobo3b2o4b2o3bo$2bo2bo2bo2bo2bo2bobo2bo2bo2b o2bo2bo$2bo2bo2bo2bo2bo2bobo2bo2bo2bo2bo2bo$2bo3b2o4b2o3bobo3b2o4b2o3b o$2bo14bobo14bo$2bo14bobo14bo$2b16ob16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+2 with two orthogonal gutters, one doubly skewed

x = 35, y = 35, rule = B/S012345678 17b16o$17bo14bo$16obo14bo$o14bobo3b2o4b2o3bo$o14bobo2bo2bo2bo2bo2bo$o 3b2o4b2o3bobo2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bobo3b2o4b2o3bo$o2bo2bo2bo 2bo2bobo14bo$o3b2o4b2o3bobo14bo$o14bobo2b10o2bo$o14bobo8b4o2bo$o2b10o 2bobo8b4o2bo$o2b4o8bobo9b2o3bo$o2b4o8bobo14bo$o3b2o9bobo14bo$o14bob16o $o14bo$16o3b16o$19bo14bo$2b16obo14bo$2bo14bobo9b2o3bo$2bo14bobo8b4o2bo $2bo3b2o9bobo8b4o2bo$2bo2b4o8bobo2b10o2bo$2bo2b4o8bobo14bo$2bo2b10o2bo bo14bo$2bo14bobo3b2o4b2o3bo$2bo14bobo2bo2bo2bo2bo2bo$2bo3b2o4b2o3bobo 2bo2bo2bo2bo2bo$2bo2bo2bo2bo2bo2bobo3b2o4b2o3bo$2bo2bo2bo2bo2bo2bobo 14bo$2bo3b2o4b2o3bobo14bo$2bo14bob16o$2bo14bo$2b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_+1 with two double-skew orthogonal gutters

x = 16, y = 16, rule = B/S012345678 b15o$obo12bo$2obo3bo5bobo$obobo10bo$o2bobo2bo3bo2bo$o3bobo2b3o3bo$o4bo bo7bo$obo3bobo6bo$o3bo2bobo5bo$o4bo2bobo4bo$o4bo3bobo3bo$o4bo4bobo2bo$ o3bo6bobobo$obo9bob2o$o12bobo$15o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 160 NOGUI ]] Diagonal gutter symmetry D2_x_gD1S0

x = 16, y = 17, rule = B/S012345678 b15o$2bo12bo$o2bo3bo5bobo$2o2bo10bo$obo2bo2bo3bo2bo$o2bo2bo2b3o3bo$o3b o2bo7bo$o4bo2bo6bo$obo3bo2bo5bo$o3bo2bo2bo4bo$o4bo2bo2bo3bo$o4bo3bo2bo 2bo$o4bo4bo2bobo$o3bo6bo2b2o$obo9bo2bo$o12bo$15o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 160 HEIGHT 160 NOGUI ]] Diagonal skewgutter symmetry D2_x_gD1S1

x = 31, y = 31, rule = B/S012345678 30o$o14bo12bobo$o14bobo5bo3bob2o$o7bo6bo10bobobo$o6bobo5bo2bo3bo2bobo 2bo$o7bo3bo2bo3b3o2bobo3bo$o11bo2bo7bobo4bo$o3bo7bo2bo6bobo3bobo$o2bob o6bo2bo5bobo2bo3bo$o3bo6bo3bo4bobo2bo4bo$o9bo4bo3bobo3bo4bo$o8bo5bo2bo bo4bo4bo$o4b4o6bobobo6bo3bo$o14b2obo9bobo$o14bobo12bo$15ob15o$o12bobo 14bo$obo9bob2o14bo$o3bo6bobobo6b4o4bo$o4bo4bobo2bo5bo8bo$o4bo3bobo3bo 4bo9bo$o4bo2bobo4bo3bo6bo3bo$o3bo2bobo5bo2bo6bobo2bo$obo3bobo6bo2bo7bo 3bo$o4bobo7bo2bo11bo$o3bobo2b3o3bo2bo3bo7bo$o2bobo2bo3bo2bo5bobo6bo$ob obo10bo6bo7bo$2obo3bo5bobo14bo$obo12bo14bo$b30o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_x1 with diagonal gutter symmetry D4_x1_gD1S0

x = 31, y = 31, rule = B/S012345678 b29o$obo12bo12bobo$2obo3b2o3b2obobo5bo3bob2o$obobo10bo10bobobo$o2bobo 2bo3bo2bo2bo3bo2bobo2bo$o3bobo2b3o3bo3b3o2bobo3bo$o4bobo7bo7bobo4bo$ob o3bobo6bo6bobo3bobo$obobo2bobo5bo5bobo2bo3bo$o4bo2bobo4bo4bobo2bo4bo$o 4bo3bobo3bo3bobo3bo4bo$o4bo4bobo2bo2bobo4bo4bo$obobo6bobobobobo6bo3bo$ obo9bob3obo9bobo$o12bobobo12bo$15ob15o$o12bobobo12bo$obo9bob3obo9bobo$ o3bo6bobobobobo6bobobo$o4bo4bobo2bo2bobo4bo4bo$o4bo3bobo3bo3bobo3bo4bo $o4bo2bobo4bo4bobo2bo4bo$o3bo2bobo5bo5bobo2bobobo$obo3bobo6bo6bobo3bob o$o4bobo7bo7bobo4bo$o3bobo2b3o3bo3b3o2bobo3bo$o2bobo2bo3bo2bo2bo3bo2bo bo2bo$obobo10bo10bobobo$2obo3bo5bobob2o3b2o3bob2o$obo12bo12bobo$b29o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0

x = 32, y = 32, rule = B/S012345678 31o$o14b2o12bobo$o14b2obo5bo3bob2o$o7bo6b2o10bobobo$o6bobo5b2o2bo3bo2b obo2bo$o7bo3bo2b2o3b3o2bobo3bo$o11bo2b2o7bobo4bo$o3bo7bo2b2o6bobo3bobo $o2bobo6bo2b2o5bobo2bo3bo$o3bo6bo3b2o4bobo2bo4bo$o9bo4b2o3bobo3bo4bo$o 8bo5b2o2bobo4bo4bo$o4b4o6b2obobo6bo3bo$o14b3obo9bobo$o14b2obo12bo$15o 2b15o$15o2b15o$o12bob2o14bo$obo9bob3o14bo$o3bo6bobob2o6b4o4bo$o4bo4bob o2b2o5bo8bo$o4bo3bobo3b2o4bo9bo$o4bo2bobo4b2o3bo6bo3bo$o3bo2bobo5b2o2b o6bobo2bo$obo3bobo6b2o2bo7bo3bo$o4bobo7b2o2bo11bo$o3bobo2b3o3b2o2bo3bo 7bo$o2bobo2bo3bo2b2o5bobo6bo$obobo10b2o6bo7bo$2obo3bo5bob2o14bo$obo12b 2o14bo$b31o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_x4 with diagonal gutter symmetry D4_x4_gD1S0

x = 32, y = 32, rule = B/S012345678 b30o$obo12b2o12bobo$2obo3bo5bob2ob2o3b2o3bob2o$obobo10b2o10bobobo$o2bo bo2bo3bo2b2o2bo3bo2bobo2bo$o3bobo2b3o3b2o3b3o2bobo3bo$o4bobo7b2o7bobo 4bo$obo3bobo6b2o6bobo3bobo$o3bo2bobo5b2o5bobo2bobobo$o4bo2bobo4b2o4bob o2bo4bo$o4bo3bobo3b2o3bobo3bo4bo$o4bo4bobo2b2o2bobo4bo4bo$o3bo6bobob2o bobo6bobobo$obo9bob4obo9bobo$o12bob2obo12bo$15o2b15o$15o2b15o$o12bob2o bo12bo$obo9bob4obo9bobo$obobo6bobob2obobo6bo3bo$o4bo4bobo2b2o2bobo4bo 4bo$o4bo3bobo3b2o3bobo3bo4bo$o4bo2bobo4b2o4bobo2bo4bo$obobo2bobo5b2o5b obo2bo3bo$obo3bobo6b2o6bobo3bobo$o4bobo7b2o7bobo4bo$o3bobo2b3o3b2o3b3o 2bobo3bo$o2bobo2bo3bo2b2o2bo3bo2bobo2bo$obobo10b2o10bobobo$2obo3b2o3b 2ob2obo5bo3bob2o$obo12b2o12bobo$b30o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0

x = 33, y = 33, rule = B/S012345678 16ob16o$o14bobo14bo$o14bobo14bo$o7bo6bobo6bo7bo$o6bobo5bobo5bobo6bo$o 7bo3bo2bobo2bo3bo7bo$o11bo2bobo2bo11bo$o3bo7bo2bobo2bo7bo3bo$o2bobo6bo 2bobo2bo6bobo2bo$o3bo6bo3bobo3bo6bo3bo$o9bo4bobo4bo9bo$o8bo5bobo5bo8bo $o4b4o6bobo6b4o4bo$o14bobo14bo$o14bobo14bo$16ob16o2$16ob16o$o14bobo14b o$o14bobo14bo$o4b4o6bobo6b4o4bo$o8bo5bobo5bo8bo$o9bo4bobo4bo9bo$o3bo6b o3bobo3bo6bo3bo$o2bobo6bo2bobo2bo6bobo2bo$o3bo7bo2bobo2bo7bo3bo$o11bo 2bobo2bo11bo$o7bo3bo2bobo2bo3bo7bo$o6bobo5bobo5bobo6bo$o7bo6bobo6bo7bo $o14bobo14bo$o14bobo14bo$16ob16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 with orthogonal gutter symmetry

x = 31, y = 31, rule = B/S012345678 b29o$obo12bo12bobo$2obo3bo5bobobo5bo3bob2o$obobo10bo10bobobo$o2bobo2bo 3bo2bo2bo3bo2bobo2bo$o3bobo2b3o3bo3b3o2bobo3bo$o4bobo7bo7bobo4bo$obo3b obo6bo6bobo3bobo$o3bo2bobo5bo5bobo2bo3bo$o4bo2bobo4bo4bobo2bo4bo$o4bo 3bobo3bo3bobo3bo4bo$o4bo4bobo2bo2bobo4bo4bo$o3bo6bobobobobo6bo3bo$obo 9bob3obo9bobo$o12bobobo12bo$15ob15o$o12bobobo12bo$obo9bob3obo9bobo$o3b o6bobobobobo6bo3bo$o4bo4bobo2bo2bobo4bo4bo$o4bo3bobo3bo3bobo3bo4bo$o4b o2bobo4bo4bobo2bo4bo$o3bo2bobo5bo5bobo2bo3bo$obo3bobo6bo6bobo3bobo$o4b obo7bo7bobo4bo$o3bobo2b3o3bo3b3o2bobo3bo$o2bobo2bo3bo2bo2bo3bo2bobo2bo $obobo10bo10bobobo$2obo3bo5bobobo5bo3bob2o$obo12bo12bobo$b29o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 with diagonal gutter symmetry

x = 33, y = 33, rule = B/S012345678 b15ob15o$obo12bobo12bobo$2obo3bo5bobobobo5bo3bob2o$obobo10bobo10bobobo $o2bobo2bo3bo2bobo2bo3bo2bobo2bo$o3bobo2b3o3bobo3b3o2bobo3bo$o4bobo7bo bo7bobo4bo$obo3bobo6bobo6bobo3bobo$o3bo2bobo5bobo5bobo2bo3bo$o4bo2bobo 4bobo4bobo2bo4bo$o4bo3bobo3bobo3bobo3bo4bo$o4bo4bobo2bobo2bobo4bo4bo$o 3bo6bobobobobobo6bo3bo$obo9bob2ob2obo9bobo$o12bobobobo12bo$15o3b15o2$ 15o3b15o$o12bobobobo12bo$obo9bob2ob2obo9bobo$o3bo6bobobobobobo6bo3bo$o 4bo4bobo2bobo2bobo4bo4bo$o4bo3bobo3bobo3bobo3bo4bo$o4bo2bobo4bobo4bobo 2bo4bo$o3bo2bobo5bobo5bobo2bo3bo$obo3bobo6bobo6bobo3bobo$o4bobo7bobo7b obo4bo$o3bobo2b3o3bobo3b3o2bobo3bo$o2bobo2bo3bo2bobo2bo3bo2bobo2bo$obo bo10bobo10bobobo$2obo3bo5bobobobo5bo3bob2o$obo12bobo12bobo$b15ob15o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 with orthogonal and diagonal gutters

x = 32, y = 32, rule = B/S012345678 b30o$obo12b2o12bobo$2obo3bo5bob2obo5bo3bob2o$obobo10b2o10bobobo$o2bobo 2bo3bo2b2o2bo3bo2bobo2bo$o3bobo2b3o3b2o3b3o2bobo3bo$o4bobo7b2o7bobo4bo $obo3bobo6b2o6bobo3bobo$o3bo2bobo5b2o5bobo2bo3bo$o4bo2bobo4b2o4bobo2bo 4bo$o4bo3bobo3b2o3bobo3bo4bo$o4bo4bobo2b2o2bobo4bo4bo$o3bo6bobob2obobo 6bo3bo$obo9bob4obo9bobo$o12bob2obo12bo$15o2b15o$15o2b15o$o12bob2obo12b o$obo9bob4obo9bobo$o3bo6bobob2obobo6bo3bo$o4bo4bobo2b2o2bobo4bo4bo$o4b o3bobo3b2o3bobo3bo4bo$o4bo2bobo4b2o4bobo2bo4bo$o3bo2bobo5b2o5bobo2bo3b o$obo3bobo6b2o6bobo3bobo$o4bobo7b2o7bobo4bo$o3bobo2b3o3b2o3b3o2bobo3bo $o2bobo2bo3bo2b2o2bo3bo2bobo2bo$obobo10b2o10bobobo$2obo3bo5bob2obo5bo 3bob2o$obo12b2o12bobo$b30o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_4 with diagonal gutter symmetry

x = 34, y = 34, rule = B/S012345678 17b16o$16obo14bo$o14bobo14bo$o14bobo6bo7bo$o7bo6bobo5bobo6bo$o6bobo5bo bo2bo3bo7bo$o7bo3bo2bobo2bo11bo$o11bo2bobo2bo7bo3bo$o3bo7bo2bobo2bo6bo bo2bo$o2bobo6bo2bobo3bo6bo3bo$o3bo6bo3bobo4bo9bo$o9bo4bobo5bo8bo$o8bo 5bobo6b4o4bo$o4b4o6bobo14bo$o14bobo14bo$o14bob16o$16o$18b16o$b16obo14b o$bo14bobo14bo$bo14bobo6b4o4bo$bo4b4o6bobo5bo8bo$bo8bo5bobo4bo9bo$bo9b o4bobo3bo6bo3bo$bo3bo6bo3bobo2bo6bobo2bo$bo2bobo6bo2bobo2bo7bo3bo$bo3b o7bo2bobo2bo11bo$bo11bo2bobo2bo3bo7bo$bo7bo3bo2bobo5bobo6bo$bo6bobo5bo bo6bo7bo$bo7bo6bobo14bo$bo14bobo14bo$bo14bob16o$b16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 with rotationally-symmetric orthogonal skewgutter symmetry

x = 34, y = 34, rule = B/S012345678 17b15o$b15obo12bobo$obo12bobobo5bo3bob2o$2obo3bo5bobobo10bobobo$obobo 10bobo2bo3bo2bobo2bo$o2bobo2bo3bo2bobo3b3o2bobo3bo$o3bobo2b3o3bobo7bob o4bo$o4bobo7bobo6bobo3bobo$obo3bobo6bobo5bobo2bo3bo$o3bo2bobo5bobo4bob o2bo4bo$o4bo2bobo4bobo3bobo3bo4bo$o4bo3bobo3bobo2bobo4bo4bo$o4bo4bobo 2bobobobo6bo3bo$o3bo6bobobob2obo9bobo$obo9bob2obobo12bo$o12bobo2b15o$ 15o$19b15o$b15o2bobo12bo$bo12bobob2obo9bobo$bobo9bob2obobobo6bo3bo$bo 3bo6bobobobo2bobo4bo4bo$bo4bo4bobo2bobo3bobo3bo4bo$bo4bo3bobo3bobo4bob o2bo4bo$bo4bo2bobo4bobo5bobo2bo3bo$bo3bo2bobo5bobo6bobo3bobo$bobo3bobo 6bobo7bobo4bo$bo4bobo7bobo3b3o2bobo3bo$bo3bobo2b3o3bobo2bo3bo2bobo2bo$ bo2bobo2bo3bo2bobo10bobobo$bobobo10bobobo5bo3bob2o$b2obo3bo5bobobo12bo bo$bobo12bob15o$2b15o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] and diagonal gutters

x = 35, y = 35, rule = B/S012345678 17b16o$17bo14bo$16obo14bo$o14bobo6bo7bo$o14bobo5bobo6bo$o7bo6bobo2bo3b o7bo$o6bobo5bobo2bo11bo$o7bo3bo2bobo2bo7bo3bo$o11bo2bobo2bo6bobo2bo$o 3bo7bo2bobo3bo6bo3bo$o2bobo6bo2bobo4bo9bo$o3bo6bo3bobo5bo8bo$o9bo4bobo 6b4o4bo$o8bo5bobo14bo$o4b4o6bobo14bo$o14bob16o$o14bo$16o3b16o$19bo14bo $2b16obo14bo$2bo14bobo6b4o4bo$2bo14bobo5bo8bo$2bo4b4o6bobo4bo9bo$2bo8b o5bobo3bo6bo3bo$2bo9bo4bobo2bo6bobo2bo$2bo3bo6bo3bobo2bo7bo3bo$2bo2bob o6bo2bobo2bo11bo$2bo3bo7bo2bobo2bo3bo7bo$2bo11bo2bobo5bobo6bo$2bo7bo3b o2bobo6bo7bo$2bo6bobo5bobo14bo$2bo7bo6bobo14bo$2bo14bob16o$2bo14bo$2b 16o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry

x = 35, y = 35, rule = B/S012345678 17b15o$17bo12bobo$b15obobo5bo3bob2o$obo12bobo10bobobo$2obo3bo5bobobo2b o3bo2bobo2bo$obobo10bobo3b3o2bobo3bo$o2bobo2bo3bo2bobo7bobo4bo$o3bobo 2b3o3bobo6bobo3bobo$o4bobo7bobo5bobo2bo3bo$obo3bobo6bobo4bobo2bo4bo$o 3bo2bobo5bobo3bobo3bo4bo$o4bo2bobo4bobo2bobo4bo4bo$o4bo3bobo3bobobobo 6bo3bo$o4bo4bobo2bob2obo9bobo$o3bo6bobobobobo12bo$obo9bob2o2b15o$o12bo bo$15o5b15o$19bobo12bo$2b15o2b2obo9bobo$2bo12bobobobobo6bo3bo$2bobo9bo b2obo2bobo4bo4bo$2bo3bo6bobobobo3bobo3bo4bo$2bo4bo4bobo2bobo4bobo2bo4b o$2bo4bo3bobo3bobo5bobo2bo3bo$2bo4bo2bobo4bobo6bobo3bobo$2bo3bo2bobo5b obo7bobo4bo$2bobo3bobo6bobo3b3o2bobo3bo$2bo4bobo7bobo2bo3bo2bobo2bo$2b o3bobo2b3o3bobo10bobobo$2bo2bobo2bo3bo2bobobo5bo3bob2o$2bobobo10bobo 12bobo$2b2obo3bo5bobob15o$2bobo12bo$3b15o! [[ THEME 6 GRID GRIDMAJOR 0 ZOOM 8 WIDTH 320 HEIGHT 320 NOGUI ]] and diagonal gutters

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.

Preserving triple or higher orthogonal skewgutter symmetry in a range-1 Moore rule requires that a pattern must not be able to escape its bounding box.

"Zero-dimensional" gutters of a single cell can also exist, and indeed do exist in Life (in which they form immediately due to the lack of S0, S4 or S8). However, these are of absolutely no interest due to applying to every pattern with a symmetry that can support them.

On a hexagonal grid

Non-trivial gutter symmetry is also known to exist on hexagonal grids. Searching with these is not yet supported by apgsearch.

Compositional/translational symmetry

In certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating what can be described as an isotropic Margolus-neighbourhood rule, or block cellular automaton. The most notable example of this is 2×2, whioch does this with solid 2×2 regions of four cells.

In the case of 2×2's rule, further self-similarity can be noted with certain patterns - anything made of solid 4×4 blocks which are correctly aligned will still be made of 4×4 blocks in even generations, and so on.

apgsearch supports the full block version of these symmetries through the inflation operator.

Most well-known wicks and agars have translational symmetry, in one and two dimensions respectively, though some may also have reflectional symmetry, and agars can have rotational symmetry.

Life exhibits this with single cells spaced out from each other - examples of oscillators where this can be seen include the barberpole.

wip below

Other conserved qualities

There are many rules, particularly those in the Margolus and Partitioned Cellular Automata rulespaces, which nontrivially conserve population, such that their evolution will never cause the population to change.

These two rulespaces also prominently feature reversible rules, in which a rule has a correspondign reverse rule which equates to the pattern evolving perfectly backwards.