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Spaceship

2026-09-06T09:22:26Z

Remu: /* In other rules */ <tab> was broken.

{{Glossary}}
[[File:Lwss.gif|thumb|right|The lightweight spaceship in [[Conway's Game of Life]]]]A '''spaceship''' (also referred to as a '''glider''',<ref>{{CiteLexicon|file=lex_g.htm#glider|name=Glider|accessdate=April 18, 2009}}</ref> or less commonly a '''fish''',<ref>{{CiteLexicon|file=lex_f.htm#fish|name=Fish|accessdate=April 18, 2009}}</ref> and commonly shortened to "ship") is a finite [[pattern]] that returns to its initial state after a number of [[generation]]s (known as its [[Period#Spaceships|period]]) but in a different location.

__TOC__

== Spaceship speed ==
{{main|Speed}}
The speed of a spaceship is the number of [[cell]]s that the pattern moves during its period, divided by the period length. This is expressed in terms of ''c'' (the metaphorical "speed of light") which is one cell per generation; thus, a spaceship with a period of five that moves two cells to the left during its period travels at the speed of [[2c/5 orthogonal|2c/5]].

Until the construction of [[Gemini]] in May {{year|2010}}, all known spaceships in [[Conway's Game of Life|Life]] traveled either orthogonally (only horizontal or vertical displacement) or diagonally (equal horizontal and vertical displacement); other [[oblique spaceship]]s have been constructed since then, e.g. [[waterbear]] and the [[Parallel HBK]]. It is known that Life has spaceships that travel in all rational directions at arbitrarily slow speeds (see [[universal constructor]]).

Spaceships traveling in other directions and at different speeds have also been constructed in other two-dimensional [[cellular automaton|cellular automata]].<ref name="fano_ca" />

The maximum speed for Life is [[c/2 orthogonal]] and [[c/4 diagonal]]<ref name="spaceship_speed_limits_in_life_like_cellular_automata" /> (and continually, (2, 1)c/6 for [[knightship|slope 2]], (3, 1)c/8 for [[slope]] 3, etc.); however, certain [[agar crawler]]s such as [[lightspeed wire]] can break this speed in their respective medium. Spaceships in the traditional [[Moore neighbourhood]] of range 1 have a maximum speed of c, although [[Larger than Life]] neighbourhoods can increase this limit depending on the interactable distance between cells. For Life, no spaceship can move at speed (''m'',''n'')c/''x'' where (''m''+''n'')/''x'' > 1/2; for other non-strobing [[Life-like cellular automaton|outer-totalistic]] and [[Non-totalistic cellular automaton|non-totalistic]] rules, the limit is at most 1, and in strobing rules, 3/2.<ref name="post56111">{{LinkForumThread|f=11|t=3193|p=56111|title=Re: B0 hyper-relativistic speeds|author=vyznev|date=February 4, 2018|format=raw}}, in which it was noted that INT rules have a speed limit of <tt>x+y ≤ {{frac|3|2}}*p</tt></ref>
Certain range-1 [[non-isotropic cellular automaton|non-isotropic]] rules can harbour [[c diagonal]] spaceships, giving a limit of 2.

== Spaceship types ==
{{main|Types of spaceships}}
Spaceships are most commonly sorted by their speed and direction, and sometimes their period. They can be separated into three fundamental categories:

=== Elementary spaceships ===
[[File:AlmostKnightship.png|thumb|right|An "[[almost knightship]]"]]
''Elementary spaceships'' are the smallest classification of spaceships. This class consists of naturally-occurring ships, as well as those found by direct computer search, such as the [[weekender]].

Despite their generally small size, only a few of them have been known to have emerged from [[soup]]s.

Constructing [[gun]]s for elementary spaceships is usually difficult, as only a few have known glider syntheses (with those that do generally requiring a lot of gliders in awkward positions), with larger elementary spaceships tending to contain high concentrations of [[space dust]].

The following table lists the smallest known cases for the lowest possible periods for each speed:

{| class="wikitable"
! Speed
! Slope
! Direction
! Smallest known
! Minimum # of [[cell]]s
|-
! [[c/2 orthogonal|c/2]]
! 0
| orthogonal
| [[64P2H1V0]]
| {{cells|64}}
|-
! [[c/3 orthogonal|c/3]]
! 0
| orthogonal
| [[25P3H1V0.1]]<ref group="n">[[25P3H1V0.2]] has the same population of 25, but has a more expansive bounding box</ref>
| {{cells|25}}
|-
! [[c/4 orthogonal|c/4]]
! 0
| orthogonal
| [[37P4H1V0]]
| {{cells|37}}
|-
! [[c/5 orthogonal|c/5]]
! 0
| orthogonal
| [[spider]]
| {{cells|58}}
|-
! [[2c/5 orthogonal|2c/5]]
! 0
| orthogonal
| [[30P5H2V0]]
| {{cells|30}}
|-
! [[c/6 orthogonal|c/6]]
! 0
| orthogonal
| [[56P6H1V0]]
| {{cells|56}}
|-
! [[c/7 orthogonal|c/7]]
! 0
| orthogonal
| [[loafer]]
| {{cells|20}}
|-
! [[2c/7 orthogonal|2c/7]]
! 0
| orthogonal
| [[weekender]]
| {{cells|36}}
|-
! [[3c/7 orthogonal|3c/7]]
! 0
| orthogonal
| [[232P7H3V0]]
| {{cells|232}}
|-
! [[c/10 orthogonal|c/10]]
! 0
| orthogonal
| [[copperhead]]
| {{cells|28}}
|-
! colspan="5" |
|-
! [[c/4 diagonal|c/4]]
! 1
| diagonal
| [[glider]]
| {{cells|5}}
|-
! [[c/5 diagonal|c/5]]
! 1
| diagonal
| [[58P5H1V1]]
| {{cells|58}}
|-
! [[c/6 diagonal|c/6]]
! 1
| diagonal
| [[77P6H1V1]]
| {{cells|77}}
|-
! [[c/7 diagonal|c/7]]
! 1
| diagonal
| [[lobster]]
| {{cells|83}}
|-
! [[c/8 diagonal|c/8]]
! 1
| diagonal
| [[walrus]]
| {{cells|68}}
|-
! colspan="5" |
|-
! [[(2,1)c/6]]
! 2
| [[knightwise]]
| [[Sir Robin]]
| {{cells|282}}
|}
<references group="n"/>

The following is an incomplete list of some higher-period ships which travel at the same velocities:<ref>{{CiteSummersPattern|name=jslife|accessdate=January 17, 2019}}</ref>
{| class="wikitable"
! Speed
! Direction
! Smallest known
! Minimum # of [[cell]]s
|-
! [[c/2 orthogonal|2c/4]]
| orthogonal
| [[lightweight spaceship]]
| {{cells|9}}
|-
! [[c/2 orthogonal|3c/6]]
| orthogonal
| [[86P6H3V0]]
| {{cells|86}}
|-
! [[c/3 orthogonal|2c/6]]
| orthogonal
| [[72P6H2V0]]
| {{cells|72}}
|-
! [[c/2 orthogonal|4c/8]]
| orthogonal
| [[31P8H4V0]]
| {{cells|31}}
|-
! [[c/4 orthogonal|2c/8]]
| orthogonal
| [[74P8H2V0]]
| {{cells|74}}
|-
! [[c/3 orthogonal|3c/9]]
| orthogonal
| [[55P9H3V0]]
| {{cells|55}}
|-
! [[c/2 orthogonal|5c/10]]
| orthogonal
| {{LinkCatagolue|xq10_wg0cc4625ggowgo8cy5gz4et915x137c4clkig0g0s6231fpogzx3l5e3y310665d0ew8owh6qaa33zx2d0ly9g0gwgo50d0vggaz0gom5536y0g0gc8bb5d0ry0gosb2245cz13d4klx4efp11dpigx132647czy0111325y41|patternname=214P10H5V0|style=raw}}
| {{cells|214}}
|-
! [[2c/5 orthogonal|4c/10]]
| orthogonal
| [[66P10H4V0]]
| {{cells|66}}
|-
! [[c/5 orthogonal|2c/10]]
| orthogonal
| {{LinkCatagolue|xq10_o8ul1431z12fe022420626c66z0rhh0rzg8ue08848z32flg4og|patternname=68P10H2V0|style=raw}}
| {{cells|68}}
|-
! [[c/2 orthogonal|6c/12]]
| orthogonal
| [[35P12H6V0]]
| {{cells|35}}
|-
! [[c/3 orthogonal|4c/12]]
| orthogonal
| {{LinkCatagolue|xq12_y6g81t0t18gzy4so4k7ece7k4oszy7l3i4i3lzy52q9fxf9q2z25c2ce057y7750ec2c52|patternname=103P12H4V0|style=raw}}
| {{cells|103}}
|-
! [[c/4 orthogonal|3c/12]]
| orthogonal
| {{LinkCatagolue|xq12_2d54031013045d2yd2d54031013045d2zvfg8sy1s8gfvy0oc2u0u2coy0vfg8sy1s8gfvzdsl211gng112lsdy15580855y1dsl211gng112lsdz058gsg131gsg85y0oo0lj0jl0ooy058gsg131gsg85zx55qxq55y2edh0hph0hdey255qxq55zy03626263y3114b616b411y33626263zyg2320232|patternname=314P12H3V0|style=raw}}
| {{cells|314}}
|-
! [[c/6 orthogonal|2c/12]]
| orthogonal
| {{LinkCatagolue|xq12_yj13922y022931zymo7q99q7ozyggg0nj611tuut116jn0ggzyi1y0gc44cgy01z03m0m3y03m0m3y8223322y93m0m3y03m0m3zw3gn3aaaa3ng3y7o035w530oy83gn3aaaa3ng3zwu97y279uy812222221y9u97y279uz22qggy2ggq220248oe2yb2eo842022qggy2ggq22zxq03q22q30qy526csssssssssssc62y5q03q22q30qzxdidy0didybe2i2eybdidy0didzyk8g40cscsc04g8zyjg0omb86068bmo0gzyj1247y37421zyk4ao9f9o9f9oa4zykg8kg0o5o0gk8gzyk12jg12121gj21zyj11cn4mo0om4nc11zyjgo0g61x16g0ogzyhegcd16b6g0g6b61dcgezyhc4g122e0a0a0e221g4czyh124epni2h0h2inpe421zyjg8mti47074itm8gzyg8u125g11gg0gg11g521u8zyj1h9o8pp0pp8o9h1zyj12t60uu0uu06t21zymora202arozyi80a191vg0gv191a08zyh18d6oy5o6d81zyk1gp4y14pg1zyhgg07g24gxg42g70ggzyh1210121y11210121|patternname=861P12H2V0|style=raw|format=extended}}
| {{cells|861}}
|-
! [[c/2 orthogonal|7c/14]]
| orthogonal
| {{LinkCatagolue|xq14_y6ogmm23zy427ekgi0gzy71q2n1zy71mgqzy5g0gr31w3mh0c0gwgxgy3gzwogmm23125dkl0eb8a9ll51x8h11y0311z27ekgi0gy1120612zx1q2n1zx1mgqzgosb2i13z0162qqggzy31|patternname=174P14H7V0|style=raw}}
| {{cells|174}}
|-
! [[c/3 orthogonal|5c/15]]
| orthogonal
| {{LinkCatagolue|xq15_sa408bb5z0344pa3t123z4ah0vwvv0a0ez0o44jaong8oy06mhgz7a402qqky1c4c111zy8br5gzya1|patternname=112P15H5V0|style=raw}}
| {{cells|112}}
|-
! [[c/5 orthogonal|3c/15]]
| orthogonal
| [[p15 pre-pulsar spaceship]]
| {{cells|252}}<ref name="post153692" />
|-
! [[c/2 orthogonal|8c/16]]
| orthogonal
| [[Coe ship]]
| {{cells|28}}
|-
! [[c/2 orthogonal|9c/18]]
| orthogonal
| {{LinkCatagolue|xq18_ykggy4g8go0v1s0gzyl364kg4li620hl0v0j220aigz59o0o4gg0th8a0sos0a8ht0gg4o0uh67y06os0v1kg880988zx1344421y71244431wggxg0ggqa1x1fc3t0g8xq2gk0ogo0kg2qx8g0giazyq24c0vge833y41127899431x111x134998721zyd8oy38o0u1coc72zye338fgps87431zyj1|patternname=293P18H9V0|style=raw|format=extended}}
| {{cells|293}}
|-
! [[c/3 orthogonal|6c/18]]
| orthogonal
| {{LinkCatagolue|xq18_061ugzw1cdcy3iccy1hmsgwa6my2ggz3jroy3gog4n081qq222q38c2agr2amf2of2z0gggx66wvvm9g609ggxg90999gg09pf0gzccd1y41g2e01855kkklc134l0d456fkhf4zw8jr3y3433y1863x566z0687|patternname=242P18H6V0|style=raw}}
| {{cells|242}}
|-
! [[c/6 orthogonal|3c/18]]
| orthogonal
| {{LinkCatagolue|xq18_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|patternname=861P18H3V0|style=raw|format=extended}}
| {{cells|861}}
|-
! [[c/2 orthogonal|10c/20]]
| orthogonal
| [[sidewinder]]
| {{cells|32}}
|-
! [[c/10 orthogonal|2c/20]]
| orthogonal
| [[Fireship|fireship tagalong]]
| {{cells|131}}
|-
! [[2c/7 orthogonal|6c/21]]
| orthogonal
| [[doo-dah]]
| {{cells|74}}
|-
! [[c/2 orthogonal|11c/22]]
| orthogonal
| {{LinkCatagolue|xq22_1fe2o2s0giay1gyfggxg0ggo8zx11avg180g4gu0749c8899988egs0g0g0g24c0vge833zy3131011y2k50t9895x234e8f0aoc72zy2ggy89l0gmd8zy377hs1u0o95w8a0u1cf1zy75fogk0oi8f211zya1|patternname=213P22H11V0|style=raw|format=extended}}
| {{cells|213}}
|-
! [[c/2 orthogonal|12c/24]]
| orthogonal
| {{LinkCatagolue|xq24_xo8ox6frcz0gh242hgz6d73037d6|patternname=39P24H12V0|style=raw}}
| {{cells|39}}
|-
! [[3c/7 orthogonal|12c/28]]
| orthogonal
| [[232P7H3V0|3c/7 wave stabilization]]
| {{cells|307}}
|-
! [[c/5 orthogonal|6c/30]]
| orthogonal
| [[pre-pulsar spaceship]]
| {{cells|128}}
|-
! [[c/2 orthogonal|18c/36]]
| orthogonal
| [[35P12H6V0]] with [[pufferfish]] [[tagalong]]
| {{cells|102}}
|-
! colspan="5" |
|-
! [[c/4 diagonal|2c/8]]
| diagonal
| [[p8 swan tagalong]]
| {{cells|117}}
|-
! [[c/4 diagonal|3c/12]]
| diagonal
| {{LinkCatagolue|xq12_ym8u0mwhhh0igxoozyfgg0ggwepg09f011whuby0562zy8o8cc46baqo046jrp6zy8g5ray34pczxc0k822ijr3x247zig831y0114oczx562|patternname=142P12H3V3|style=raw|format=extended}}
| {{cells|142}}
|}

=== Engineered spaceships ===
''Engineered spaceships'' are defined as spaceships consisting of small interacting components.<ref name="new_spaceship_speed_in_conways_game_of_life" /> Some engineered spaceships are composed of hundreds of thousands or even millions of active cells; these are also sometimes referred to as ''caterpillars'' after the first engineered [[macro-spaceship]], although many 2020s developments produced results orders of magnitude smaller. Other simpler mechanisms like the [[Cordership]]s might need a relatively small number of components, down to a minimum of just two for the [[2-engine Cordership]]. Such spaceships have occurred [[natural]]ly in other rules, but not yet in [[Conway's Game of Life|Conway Life]].

Engineered spaceships have fixed speeds, because their mechanisms depend on supporting a specific active reaction that travels at one particular speed. Some rely on "[[crawler]]s", reactions in which a pattern reacts with another pattern, producing both the patterns in different positions as in the [[pi crawler]]. Others, like the [[Cordership]]s and [[pufferfish spaceship]]s, rely on ''stabilised [[puffer]] engines'', and are often referred to as "Corderoids". The process of stabilising a moving engine is referred to as [[corderisation]].

{| class="wikitable"
! Speed
! Direction
! Smallest known
! Minimum # of [[cell]]s
! Classification
|-
! [[17c/45 orthogonal|17c/45]]
| orthogonal
| [[caterpillar]]
| {{cells|11,880,039}}
| Crawler-based
|-
! [[31c/240 orthogonal|31c/240]]
| orthogonal
| [[leaf bug 2]]
| {{cells|26,571}}
| Crawler-based
|-
! [[:Category:Spaceships with speed (23,5)c/79|(23,5)c/79]]
| slope 23/5
| [[waterbear]]
| {{cells|197,896}}
| Crawler-based
|-
! [[:Category:Spaceships with speed (34,7)c/156|(34,7)c/156]]
| slope 34/7
| [[Much smaller (34,7)c/156 spaceship]]
| {{cells|647,595}}
| Crawler-based
|-
! [[:Category:Spaceships with speed (6,3)c/1024|(6,3)c/1024]]
| knightwise
| [[HBK caterpillar]]
| {{cells|329,071}}
| Crawler-based
|-
! [[:Category:Spaceships with unsimplified speed (12,4)c/62|(12,4)c/62]]
| camelwise
| [[solifuge]]
| {{cells|1,414}}
| Crawler-based
|-
! [[:Category:Spaceships with unsimplified speed (15,11)c/107|(15,11)c/107]]
| slope 15/11
| [[Silk Moth]]
| {{cells|1,190,163}}
| Crawler-based
|-
! [[:Category:Spaceships with unsimplified speed (28,3)c/84|(28,3)c/84]]
| slope 28/3
| [[Annelid]]
| {{cells|186,807,151}}
| Crawler-based
|-
! colspan="5" |
|-
! [[c/2 orthogonal|c/2]]
| orthogonal
| [[pufferfish spaceship]]
| {{cells|235}}
| Stabilized puffer engine
|-
! [[c/12 diagonal|c/12]]
| diagonal
| [[2-engine Cordership]]
| {{cells|100}}
| Stabilized puffer engine
|}

==== Traveling signal loops ====
A traveling [[signal]] loop is a subcategory somewhat similar to an engineered spaceship, but with key differences. The usual reason for constructing a traveling signal loop is to extract an extra output glider at some point in each cycle, producing an [[engineless]] [[rake]]. Suppressing a rake's output glider will trivially produce a high-period spaceship.

One main difference between engineered spaceships and traveling signal loops is that the "specific active reactions that travel at one particular speed" are spaceships themselves, and don't need any additional support. The signals traveling in the signal loops can generally be removed without having any effect on the fleet of spaceships supporting the loop.

{| class="wikitable"
! Speed
! Direction
! Examples
! Classification
|-
! [[2c/7 orthogonal|2c/7]]
| orthogonal
| [[weekender distaff]]
| High-period signal loop
|-
! [[c/3 orthogonal|c/3]]
| orthogonal
| c/3 orthogonal rake
| High-period signal loop
|-
! [[c/4 orthogonal|c/4]]
| orthogonal
| p388 c/4 forward rake, David Bell's adjustable-period backrake
| High-period signal loop
|-
! [[c/5 orthogonal|c/5]]
| orthogonal
| c/5 orthogonal rake
| High-period signal loop
|-
! [[2c/5 orthogonal|2c/5]]
| orthogonal
| 2c/5 orthogonal rake
| High-period signal loop
|-
! [[c/4 diagonal|c/4]]
| diagonal
| c/4 diagonal rake
| High-period signal loop
|-
! [[c/5 diagonal|c/5]]
| diagonal
| [[c/5 diagonal rake]]
| High-period signal loop
|-
! [[c/12 diagonal|c/12]]
| diagonal
| c/12 diagonal rake
| High-period signal loop
|}

=== Adjustable spaceships ===
''Adjustable spaceships'' (formerly known as ''engineerable spaceships'') are the third extant class of spaceships. On average smaller than the engineered spaceships in terms of population, but much larger in bounding box, their magic comes from having (to an extent) adjustable features, usually speed. With some almost always trivial modifications, these spaceships can be made to travel at different velocities and even directions. Rather than being searched for, programs exist that explicitly construct the spaceships. Adjustable spaceships can be based on variable-speed reactions such as the half-bakery and glider reaction or the freezing/reanimation cycle of the Caterloopillar, or by reading instruction tapes as in the Gemini. Families of adjustable spaceships include the Geminoids, Demonoids, Orthogonoids, loopships, half-baked knightships and Caterloopillars.

{| class="wikitable"
! Speed
! Direction
! Smallest known
! Minimum # of [[cell]]s
|-
! All speeds slower than c/4
| orthogonal
| [[Caterloopillar]] variants
| [[:Category:Patterns with (number) cells|Varies]]
|-
! 16c/217251
| orthogonal
| [[Orthogonoid]]
| {{cells|467,746}}
|-
! [[c/16384 orthogonal|c/16384]]
| orthogonal
| [[Hashlife-friendly Orthogonoid]]
| {{cells|467,585}}
|-
! [[c/32768 orthogonal|c/32768]]
| orthogonal
| [[Double-wide Orthogonoid]]
| {{cells|467,346}}
|-
! 1000130c/20003511
| orthogonal
| [[loopship]]
| {{cells|267,672}}
|-
! colspan="5" |
|-
! (200,7)c/81976184
| slope 200/7
| [[Stable Storage Spaceship]]
| {{cells|633,388}}
|-
! (5120,1024)c/16849793
| slope 5 (ibiswise)
| [[Gemini]]
| {{cells|846,278}}
|-
! (3,1)c/3948264
| slope 3 (camelwise)
| [https://conwaylife.com/forums/viewtopic.php?p=66297#p66297 camelship]
| {{cells|239,822}}
|-
! (4096,2048)c/17783745
| slope 2 (knightwise)
| [https://conwaylife.com/forums/viewtopic.php?f=2&t=406&start=25#p2578 Gemini 2]
| [[:Category:Patterns with between 100,000 and 999,999 cells|<=872,252]]
|-
! (6,3)c/245912
| slope 2 (knightwise)
| [[Parallel HBK]]
| {{cells|132,945}}
|-
! (6,3)c/2621440
| slope 2 (knightwise)
| [[Half-baked knightship]]
| {{cells|1,049,395}}
|-
! colspan="5" |
|-
! 65c/438852
| diagonal
| [[Demonoid|0hd Demonoid]]
| {{cells|27,250}}
|-
! 65c/818356
| diagonal
| [[Demonoid|10hd Demonoid]]
| {{cells|47,701}}
|-
! 79c/1183842
| diagonal
| [[Demonoid|single-channel Demonoid]]
| {{cells|60,672}}
|-
! c/16384
| diagonal
| [[Demonoid|Scorbie's Demonoid]]
| {{cells|72,207}}
|-
! c/512
| diagonal
| [[Demonoid|Hashlife-friendly Demonoid]]
| {{cells|105,369}}
|-
! 511c/1677216
| diagonal
| [[Thessalonic Demonoid]]
| {{cells|328,699}}
|-
! 1746011c/8916896
| diagonal
| [[Demonoid|Speed Demonoid]]
| {{cells|1,021,124}}
|}

=== Other classifications ===
{{main|Types of spaceships|Spaceship types}}
[[File:Greyship2 H Holzwart.png|thumb|right|A greyship found by [[Hartmut Holzwart]]]]
Although spaceships are most commonly categorized by their speed and direction, other categorizations have been applied to spaceships based on their appearances, components, or other properties. One such categorization is the symmetry of spaceships: spaceships can be bilaterally symmetric (e.g. copperhead), exhibit [[Types of spaceships#Glide symmetric spaceship|glide symmetry]] (e.g. glider), or simply be asymmetric (e.g. loafer). Other somewhat subjective categorizations have also been made, such as [[greyship]]s, spaceships filled with large amounts of static, live cells in the form of an [[agar]], or [[Types of spaceships#Smoking ship|smoking ships]], which produce large [[spark]]s, a notable example being the [[Schick engine]]. A spaceship may also support other components which would not function as spaceships on their own. Given a freestanding spaceship, such additional components are often referred to as [[tagalong]]s; however they can be attached to any side of a spaceship, such as [[pushalong 1]] and [[sidecar]]. Unstable spaceships immersed in a sustaining cloud are known as [[Types of spaceships#Flotilla|flotillae]]. A well known example is that of the [[overweight spaceship]]s, which are unstable alone but may be 'escorted' by two or more smaller spaceships.

==History==

===1970s===
[[File:mwss.gif|thumb|right|A middleweight spaceship]]
The four smallest spaceships in life, the glider, lightweight spaceship, [[middleweight spaceship]] and [[heavyweight spaceship]], were all found by hand in {{year|1970}}. For almost twenty years spaceship development was limited to adding [[tagalong]]s to the known [[c/2 orthogonal|c/2]] spaceships, such as the Schick engine.

===1980s===
Significant advances in spaceship technology came in {{year|1989}}, when [[Dean Hickerson]] began using [[Computer search|automated searches]] based on a depth-first backtracking algorithm. These searches found orthogonal spaceships with speeds of [[c/3 orthogonal|c/3]] and [[c/4 orthogonal|c/4]], new [[c/2 orthogonal|c/2]] ships, and the first spaceship other than the glider to travel at the speed of [[c/4 diagonal]]ly, dubbed the [[big glider]].

===1990s===
Hickerson continued to find new spaceship speeds, the first of this decade being [[2c/5 orthogonal]], plus several ways to combine [[switch engine]]s to create the first [[c/12 diagonal]] spaceships, named [[Cordership]]s in honour of [[Charles Corderman]].

The next spaceship speed to be discovered was that of the orthogonal [[c/5 orthogonal|c/5]] [[snail]], found by [[Tim Coe]] in {{year|1996}}, with a program he had designed using breadth-first searching, and which could split tasks between multiple CPUs.<ref name="lifepage_patterns_p5ss" /> In the following year, [[David Bell]] found the much smaller c/5 [[spider]] using [[lifesrc]], a program based on Hickerson's search algorithm.<ref name="lifepage_patterns_newc5" />

In March of {{year|1998}} [[David Eppstein]] created [[gfind]], a breadth-first program that uses a depth-first search to limit the size of the search queue.<ref name="arxiv_0004003v2" />

===2000s===
[[File:Weekender small.gif|thumb|right|David Eppstein's weekender]]
Eppstein put his search program to good use in {{year|2000}}, discovering the first spaceship that travels at the speed of [[2c/7 orthogonal]]ly, the [[weekender]]. A search by [[Paul Tooke]] using the same program found the first [[c/6 orthogonal]] spaceship, the [[dragon]], later that year. Also in 2000, [[Jason Summers]] found the first [[c/5 diagonal]] spaceship using David Bell's lifesrc program.

In {{year|2004}} [[Gabriel Nivasch]], with the help of Jason Summers and David Bell, finished construction on the [[caterpillar]], the first known orthogonal [[17c/45 orthogonal|17c/45]] spaceship, which made use of the [[17c/45 reaction]].

===2010s===
[[File:Loafer.gif|thumb|right|Josh Ball's loafer]]
In May {{year|2010}} [[Andrew J. Wade]] created a universal constructor-based spaceship, [[Gemini]], which travels at a speed of (5120,1024)c/33699586.<ref>{{LinkGoLNews|2010/05/oblique_life_spaceship_created.html|title=Oblique Life spaceship created|date=May 19, 2010|author=Adam P. Goucher}}</ref> This was the first explicitly constructed spaceship in Life to travel in an [[Types of spaceships#Knightship|oblique]] direction, and also yielded the first explicit method of constructing arbitrarily slow spaceships.

In August {{year|2011}}, Matthias Merzenich discovered [[lobster]], the first c/7 diagonal spaceship.

In February 2013, the first c/7 orthogonal spaceship, [[loafer]], was discovered by Josh Ball.

2014 provided a handful of new engineered spaceships, using various new technologies. In July, several [[half-baked knightship|half-bakery-based knightships]] were constructed with a new technique not requiring universal-constructor circuitry. These produced spaceships that were both much slower and much smaller than the Gemini variants. In September, [[Dave Greene]] and Chris Cain completed two [[31c/240 orthogonal]] spaceships, along the same general lines as the original Caterpillar but using a number of new mechanisms. Finally, in December, an oblique caterpillar dubbed the [[Waterbear]] was completed by Brett Berger, traveling at (23,5)c/79. Richard Schank discovered [[pufferfish]], a c/2 puffer, and Ivan Fomichev found a c/2 fuse for its exhaust and combined two pufferfish with fuses to assemble the first wholly high-period c/2 spaceship.

In December 2015, Chris Cain completed a diagonal self-constructing spaceship -- a [[Demonoid|"0hd Demonoid"]] -- based on Geminoid technology, adapted from a larger 10hd version constructed in November in collaboration with Dave Greene.

[[File:Copperhead.gif|framed|right|The copperhead spaceship.]]
In March 2016, [[zdr]] discovered [[copperhead]], an extremely small c/10 spaceship. A [[Fireship|pseudo-tagalong]] for this spaceship, alongside many other c/10 technologies, were constructed within two months after the discovery of the ship.

In April 2016, Michael Simkin finished the adjustable [[caterloopillar]] project, making it possible to build spaceships of arbitrary orthogonal speeds slower than c/4.

In June 2016, Tim Coe found a large elementary 3c/7 orthogonal spaceship, the [[spaghetti monster]].

[[File:Sirrobin tiny.png|thumb|right|[[Sir Robin]], the first elementary knightship to be found.<br />{{JavaRLE|sirrobin}}]]
In March 2018, [[Adam P. Goucher]] and [[Tomas Rokicki]] found the first elementary knightship, [[Sir Robin]], which translates itself by (2,1) every six generations.

===2020s===

Over the course of December 2020 and the first half of 2021, three speeds received versatile new technology, including the first intermediate-period 2c/7 orthogonal ships (based on the [[doo-dah]] by [[John Winston Garth]]), several smaller 3c/7 orthogonal ships by [[Dylan Chen]] allowing higher-period ships to be constructed, and higher-period (2,1)c/6 knightships based on the [[sprayer]] by Adam P. Goucher and Dylan Chen, all aided by the new [[ikpx2]] spaceship search program.

In terms of novel elementary speeds, the first elementary [[c/8 diagonal]] spaceship, the [[Walrus]], was found in 2023.

Other notable discoveries in the engineered spaceship area include a surprisingly small engineered camelship ([[Solifuge]]) in November 2025 based on the p124 LOM hassler reaction, several major reductions to [[31c/240]] orthogonal technology from 2024 to 2025 to the point that several different mechanisms could fit into LifeViewer windows (albeit with modifications), completions to [[(34,7)c/156]], [[(15,11)c/107]], and [[(28,3)c/184]] climbers. Additionally, various adjustable ships like a [[Thessalonic Demonoid]], a different [[Demonoid]] which fits into a LV window, and the potentially adjustable oblique ship [[Giraffe Weevil]] have been discovered.

{| class="wikitable"
|-
! Speed
! Direction
! First discovered
! Discoverer
! Year of discovery
|-
! c/4
| diagonal
| [[glider]]
| [[Richard Guy]]
| {{year|1969}}
|-
! c/2
| orthogonal
| [[lightweight spaceship]]
| [[John Conway]]
| {{year|1970}}
|-
! c/3
| orthogonal
| [[84P3H1V0.1]]
| [[Dean Hickerson]]
| {{year|1989}}
|-
! c/4
| orthogonal
| [[119P4H1V0]]
| [[Dean Hickerson]]
| {{year|1989}}
|-
! c/12
| diagonal
| [[13-engine Cordership]]
| [[Dean Hickerson]]
| {{year|1991}}
|-
! 2c/5
| orthogonal
| [[44P5H2V0]]
| [[Dean Hickerson]]
| {{year|1991}}
|-
! c/5
| orthogonal
| [[snail]]
| [[Tim Coe]]
| {{year|1996}}
|-
! 2c/7
| orthogonal
| [[weekender]]
| [[David Eppstein]]
| {{year|2000}}
|-
! c/6
| orthogonal
| [[dragon]]
| [[Paul Tooke]]
| {{year|2000}}
|-
! c/5
| diagonal
| [[295P5H1V1]]
| [[Jason Summers]]
| {{year|2000}}
|-
! 17c/45
| orthogonal
| [[caterpillar]]
| [[Gabriel Nivasch]], [[Jason Summers]], and [[David Bell]]
| {{year|2004}}
|-
! c/6
| diagonal
| [[seal]]
| [[Nicolay Beluchenko]]
| {{year|2005}}
|-
! (5120,1024)c/16849793
| slope 5 (ibiswise)
| [[Gemini]]
| [[Andrew J. Wade]]
| {{year|2010}}
|-
! (4096,2048)c/17783745
| slope 2 (knightwise)
| [https://conwaylife.com/forums/viewtopic.php?f=2&t=406&start=25#p2578 Gemini 2]
| [[Dave Greene]]
| {{year|2010}}
|-
! (3072,1024)c/36000001
| slope 3 (camelwise)
| [https://conwaylife.com/forums/viewtopic.php?f=2&t=406&start=25#p2590 Gemini 3]
| [[Dave Greene]]
| {{year|2010}}
|-
! c/7
| diagonal
| [[Lobster]]
| [[Matthias Merzenich]]
| {{year|2011}}
|-
! c/7
| orthogonal
| [[loafer]]
| [[Josh Ball]]
| {{year|2013}}
|-
! (6,3)c/2621440
| slope 2 (knightwise)
| [[Half-baked knightship]]
| [[Adam P. Goucher]]
| {{year|2014}}
|-
! (6,3)c/245912
| slope 2 (knightwise)
| [[Parallel HBK]]
| [[Chris Cain]]
| {{year|2014}}
|-
! 31c/240
| orthogonal
| [[shield bug]]
| [[Dave Greene]]
| {{year|2014}}
|-
! (23,5)c/79
| slope 23/5
| [[waterbear]]
| [[Brett Berger]]
| {{year|2014}}
|-
! 130c/(1636712+16N)
| diagonal
| [[Demonoid|10hd Demonoid]]
| [[Chris Cain]] and [[Dave Greene]]
| {{year|2015}}
|-
! 65c/(438852+16N)
| diagonal
| [[Demonoid|0hd Demonoid]]
| [[Chris Cain]]
| {{year|2015}}
|-
! c/10
| orthogonal
| [[Copperhead‎‎]]
| [[zdr]]
| {{year|2016}}
|-
! All speeds slower than c/4
| orthogonal
| [[caterloopillar]]
| [[Michael Simkin]]
| {{year|2016}}
|-
! 3c/7
| orthogonal
| [[Spaghetti monster]]
| [[Tim Coe]]
| {{year|2016}}
|-
! (2,1)c/6
| slope 2 (knightwise)
| [[Sir Robin]]
| [[Adam P. Goucher]] and [[Tomas Rokicki]]
| {{year|2018}}
|-
! All speeds slower than c/4
| diagonal
| [[Speed Demonoid]]
| [[Pavel Grankovskiy]] and [[Dave Greene]]
| {{year|2020}}
|-
! All speeds slower than c/2
| orthogonal
| [[Speed Orthogonoid]]
| [[Goldtiger997]]
| {{year|2022}}
|-
! c/8
| diagonal
| [[Walrus]]
| [[Adam P. Goucher]]
| {{year|2023}}
|-
! (34,7)c/156
| slope (34,7) (oblique)
| [[unnamed (34,7)c/156 spaceship]]
| [[Luka Okanishi]]
| {{year|2023}}
|-
! (6,2)c/31
| slope 3 (camelship)
| [[Solifuge]]
| [[Nora Brown]]
| {{year|2025}}
|-
! (15,11)c/107
|
| [[silk moth]]
| [[Goldtiger997]]
| {{year|2026}}
|-
! (28,3)c/84
|
| [[Annelid]]
| [[Brithos]]
| {{year|2026}}
|}

The first known cases for unsimplified speeds are unknown.

== In other rules ==
{| class="infobox"
|-
|[[File:OT_map_(non-Gray).png|thumb|right|A map of outer-totalistic rules. Birth conditions iterate in binary over the x axis (the right half has B0), survival in the y axis (the bottom half has S8). Outside of the bottom-left quadrant (where only the all-off agar is considered space), it is symmetrical due to the duality of [[black/white reversal]]. Green rules have known explicit examples of spaceships within them, red ones have been proven not to support any. (See file description for more details.)|256px]][[File:OT_spaceship_map.png|thumb|right|A map of outer-totalistic rules. Birth conditions iterate in a {{LinkWikipedia|Gray_code|name=Gray code|nobullet=no|nosuffix=no}} over the x axis, survival in the y axis. Green rules have known explicit examples of spaceships within them, red ones have been proven not to support any. (See file description for more details.)|256px]]

|}
Many [[Life-like cellular automata]] afford spaceships, as do their generalizations; this includes both [[outer-totalistic]] and [[isotropic non-totalistic]] rules, as well as [[non-isotropic]], [[Generations]] and [[Larger than Life]] rules.

Various segments of the Life-like [[rulespace]] cannot contain spaceships, however. For instance, assuming [[Birth on zero neighbours|B0]] is not active:

* In any rule with B1c, any pattern expands in all directions.
:* For totalistic rules, this is rounded up to B1.
* In any rule with all of S012acek3aijn4a (all survival conditions with a subset of the on cells in 4a), the [[trailing edge]] of a pattern cannot die.
:* For totalistic rules, this is rounded up to S01234.
* In any rule with B23/S0, the trailing edge of a pattern cannot die.
:*This is not the case for rules with B23 without S0; 12 of these have known [[photon]]s, and no slower spaceships may exist.<ref>{{LinkForumThread|f=11|t=5471|p=165127|title=Re: Speed limits and theorems about the existence of periodic objects|author=LaundryPizza03|date=August 15, 2023|format=ref}}</ref>
* In any rule with none of B1ce2ac3i, no pattern can escape its initial bounding box.
* In any rule with none of B1ce2ae3a, no pattern can escape its initial bounding diamond.
:* For totalistic rules, these are both rounded up to B123.
:* With B3 and none of B1245/S012345, with a pattern's bounding diamond replaced with that of its bounding box, it remains inescapable.<ref>David Eppstein, comment in [https://www.ics.uci.edu/~eppstein/ca/glider.c <tt>glider.c</tt>]</ref>
*It appears that in some rules patterns cannot escape the bounding box or the bounding diamond despite having the B3a or B2e condition. This is likely the case for {{rl|LongLife}} and {{rl|Assimilation}}.

In strobing rules with <tt>(not B1c) XOR S7c</tt> (such that one member of the pair of rules they emulate alternating between has B1c), a spaceship may move with velocity vector <tt>{{frac|(x,y)c|p}}</tt> only if <tt>x+y ≤ {{frac|3|2}}*p</tt>.<ref name="post56111" />

In rules with both of B3ai and none of B12ace, a spaceship may move at <tt>{{frac|(x,y)c|p}}</tt> only if <tt>2*max(x,y)+min(x,y) ≤ p</tt>.

In the subset without either S4w or S5a (such as Life), this criterion restricts to <tt>2*(x+y) ≤ p</tt>; the diagonal speed limit is (1,1)c/4, not (1,1)c/3, and the knightwise speed limit is (2,1)c/6, not (2,1)c/5 and so on. All patterns at the maximum speed in any direction travel at c/2 in the {{LinkWikipedia|Taxicab_metric|name=taxicab metric|nobullet=no|nosuffix=no}}.<ref name="spaceship_speed_limits_in_life_like_cellular_automata" />

The orthogonal limit without any of B1ec2a is c/2.

While assuming B0 is active, we can treat them as either [[strobing]] rules or [[black/white reversal]]s of rules. Over the outer-totalistics, this yields the following spaceshipless rulespaces:<ref>{{LinkForumThread
|format=ref|f=11|t=1074|p=93966|title=Re: Spaceships in Life-like cellular automata|author=LaundryPizza03|date=April 10, 2020}}</ref>
* Rules containing all of the transitions in at least one of {B0/S7,B08/S56,B05678/S6,B045678} without any of the transitions in at least one of {S01234,B3/S0123,B2/S0123,B23/S0,B1}
* Rules with B0234567 without S123456
Also:
* With B0/S6 and without either of B1/B2, or with B0/S67 without B2, all patterns expand at (2,1)c/2.
* With B0123 and without any of S567, patterns can never leave their bounding box.

The slowest known orthogonal elementary non-adjustable spaceship in any [[Life-like cellular automaton|Life-like rule]] is a c/5648 orthogonal spaceship in the rule {{rl|Gems}}, followed by a c/2068 orthogonal spaceship in {{rl|Gems Minor}}.
The slowest known (and highest-period) such spaceship in a range-1 [[isotropic non-totalistic rule]] is a 14c/63503454 ship in B2n3-ce4e5iy6ae/S2aei3cejk4eijkrw5cnry6-e78.<ref>{{LinkForumThread|f=11|t=6649|p=193960|title=Re: Foam Rules|author=NimbleRogue|date=September 12, 2024|format=raw}}, edit 2</ref>

=== Currently known speeds ===
There is currently an ongoing tabulation at the 5s project<ref>{{cite web|url=https://github.com/AforAmpere/sssss/tree/master/data|title=AforAmpere/sssss/tree/master/data}}</ref> cataloging the smallest known spaceships for each speed across different rules.

Alongside these, certain "series" of speeds can be proved to all exist in range-1 isotropic cellular automata:

* All orthogonal spaceships with the unsimplified speed c/n are known to exist; true-period [[Photon|c/1]], [[64P2H1V0|c/2]] and [[25P3H1V0.2|c/3]] spaceships are known.

* All orthogonal spaceships with the unsimplified speed mc/n are known to exist where n/m ≥ 9, in B2ck3ae4arw5n6k7c/S1e2-cn3aekry4-cnqw5jnry.<ref>{{LinkForumThread|f=11|t=3081|p=139574|title=Re: Re: Rules with small adjustable spaceships|author=silversmith|date=December 31, 2021|format=ref}}</ref>

==See also==
===General===
* [[Spaceship flotilla]]
* [[Spaceship terminology]]
* [[Types of spaceships]]
===Conway's Life===
* [[List of spaceships]]
* [[List of natural spaceships]]
* Subsections of [[LifeWiki:Game of Life Status page]], including [[LifeWiki:Game of Life Status page#Velocities of spaceships|velocities]] and [[LifeWiki:Game of Life Status page#Spaceship periods|periods]]
* [[LifeWiki:Spaceship Search Status Page]]
* {{CategoryLink|Spaceships}}
===Other cellular automata===
* [[Day_%26_Night#Spaceships|Day & Night#Spaceships]]
* [[HighLife#Spaceships]]
* {{rl|Grounded Life#Spaceships}}
* [[LifeWiki:Smallest Spaceships Supporting Specific Speeds]]

==References==
<references>
<ref name="arxiv_0004003v2">{{cite web
|url = https://arxiv.org/abs/cs/0004003v2
|title = Searching for Spaceships (PDF)
|author = David Eppstein
|date = April 26, 2000
|accessdate = December 4, 2023
}}</ref>
<ref name="fano_ca">{{cite web
|url = https://web.archive.org/web/20190408101245/http://fano.ics.uci.edu/ca/
|title = Gliders in Life-Like Cellular Automata
|publisher = David Eppstein
|accessdate = April 18, 2009
}}</ref>
<ref name="lifepage_patterns_newc5">{{cite web
|url = https://conwaylife.com/ref/lifepage/patterns/newc5/newc5.html
|title = New c/5 spaceship
|author = David Bell
|publisher = Paul's Page of Conway's Life Miscellany
|accessdate = April 18, 2009
}}</ref>
<ref name="lifepage_patterns_p5ss">{{cite web
|url = https://conwaylife.com/ref/lifepage/patterns/p5ss.html
|title = c/5 Orthogonal spaceship
|author = Tim Coe
|publisher = Paul's Page of Conway's Life Miscellany
|accessdate = April 18, 2009
}}</ref>
<ref name="new_spaceship_speed_in_conways_game_of_life">{{cite web
|url = https://niginsblog.wordpress.com/2016/03/07/new-spaceship-speed-in-conways-game-of-life/
|title = New Spaceship Speed in Conway’s Game of Life
|author = Alexey Nigin
|date = March 7, 2016
|accessdate = June 11, 2016
}}</ref>
<ref name="spaceship_speed_limits_in_life_like_cellular_automata">{{cite web
|url = http://www.njohnston.ca/2009/10/spaceship-speed-limits-in-life-like-cellular-automata/
|title = Spaceship Speed Limits in "B3" Life-Like Cellular Automata
|author = Nathaniel Johnston
|date = October 30, 2009
|accessdate = December 12, 2016
}}</ref>
<ref name="post153692">{{LinkForumThread
|format = ref
|p = 153692
|title = Re: Suggested LifeWiki edits
|author = Bullet51
|date = November 17, 2022
}}</ref>
</references>

==Further reading==
* [[Carter Bays]], ''Gliders in Cellular Automata'', in: Robert A. Meyers (ed). ''Encyclopedia of Complexity and Systems Science,'' Springer 2009, pp. 4240–4249.

==External links==
* {{LinkWikipedia|Spaceship_(cellular_automaton)}}
* {{LinkLexicon|filename=lex_s.htm#spaceship|patternname=Spaceship}}
* {{LinkForumThread|f=2|t=2031|title=Spaceship Discussion Thread}} (For Conway's Life)
* {{LinkForumThread|f=11|t=1074|title=Spaceships in Life-like cellular automata}} (For outer-totalistic rules)
* {{LinkForumThread|f=11|t=5175|title=Miscellaneous Spaceship (collections) in Other Cellular Automata}}

Sawtooth spaceship

2026-09-01T06:10:14Z

Remu: Add speed table

{{Glossary}}
A '''sawtooth spaceship''' is an [[pattern]] that moves like a [[spaceship]], but with a population that reaches new heights infinitely often, but also infinitely often drops below some fixed value. In other words, it is a [[sawtooth]] that leaves any area forever and also moves in the same direction as a whole, as opposed to a [[expanding sawtooth]], which does not have this requirement.<ref name="post213184" /><ref name="post213206" /> They can also be considered as [[growing spaceship]]s with a sawtooth population.<ref name="post213193" /> This term was invented by [[David Bell]] on June 12, {{year|2025}}. <ref name="post213184" />
<br>
<br>
{{EmbedViewer
|rle = x = 157, y = 75, rule = B3/S23
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|viewerconfig = #C [[ THUMBSIZE 2 ZOOM 8 WIDTH 1300 HEIGHT 700 GRID POPUPWIDTH 1400 POPUPHEIGHT 650 ]]
|position = center
|caption = Moving sawtooth 707, an example of a moving sawtooth.
|style = width:300px;
}}

==History==
The first known sawtooth spaceship was the [[moving sawtooth]], which was constructed on July 10, {{year|2005}}, by [[David Bell]]. A smaller moving sawtooth, [[moving sawtooth 707]] was constructed by [[cloudy197]] on 22 April, {{year|2011}}. <ref name="post4578" /> It was later optimized to create [[Moving sawtooth 690]] by David Bell on July 30, {{year|2025}}. It has the least infinitely repeating population of any known sawtooth spaceship.<ref name="post5225" />

==Speeds==
{| class="wikitable sortable"
! Name !! Direction !! Frontend speed !! Backend speed
|-
| [[Moving sawtooth]] || Orthogonal || [[c/2 orthogonal]] || [[c/3 orthogonal]]
|-
|}

==See also==
* [[Growing/shrinking spaceship]]
* {{CategoryLink|Sawtooth spaceships}}

== References ==
<references>
<ref name="post213184">{{LinkForumThread
|format = ref
|p = 213184
|title = Re: Thread for advanced questions
|author = David Bell
|date = June 12, 2025
}}</ref>
<ref name="post213193">{{LinkForumThread
|format = ref
|p = 213193
|title = Re: Thread for advanced questions
|author = Dave Greene
|date = June 13, 2025
}}</ref>
<ref name="post213206">{{LinkForumThread
|format = ref
|p = 213206
|title = Re: Thread for advanced questions
|author = Dave Greene
|date = June 13, 2025
}}</ref>
<ref name="post4578">{{LinkForumThread
|format = ref
|p = 4578
|title = Smaller moving sawtooth
|author = cloudy197
|date = April 22, 2011
}}</ref>
<ref name="post5225">{{LinkForumThread
|format = ref
|p = 5225
|title = Re: Smaller moving sawtooth
|author = Wojowu
|date = October 7, 2011
}}</ref>
</references>
[[Category:Everything else]]

LifeWiki:OCAFA

2026-08-31T06:03:15Z

Remu: Fix

{{#switch: {{CURRENTWEEK}}
| 21 = {{ShowFeaturedArticle|Pedestrian Life}}
| 22 = {{ShowFeaturedArticle|HighLife}}
| 23 = {{ShowFeaturedArticle|2×2}}
| 24 = {{ShowFeaturedArticle|Brian's Brain}}
| 25 = {{ShowFeaturedArticle|Pedestrian Life}}
| 26 = {{ShowFeaturedArticle|HighLife}}
| 27 = {{ShowFeaturedArticle|2×2}}
| 28 = {{ShowFeaturedArticle|Brian's Brain}}
| 29 = {{ShowFeaturedArticle|Pedestrian Life}}
| 30 = {{ShowFeaturedArticle|HighLife}}
| 31 = {{ShowFeaturedArticle|2×2}}
| 32 = {{ShowFeaturedArticle|Brian's Brain}}
| 33 = {{ShowFeaturedArticle|Pedestrian Life}}
| 34 = {{ShowFeaturedArticle|HighLife}}
| 35 = {{ShowFeaturedArticle|2×2}}
| 36 = {{ShowFeaturedArticle|Brian's Brain}}
| 37 = {{ShowFeaturedArticle|Pedestrian Life}}
| 38 = {{ShowFeaturedArticle|HighLife}}
| 39 = {{ShowFeaturedArticle|2×2}}
}}

Category:NNlk05's Rules

2026-08-30T08:49:06Z

Remu: categorising the category

Rules by NNlk05
[[Category:Alien rules]]

Category:Pages under revision

2026-08-30T08:46:38Z

Remu: categorising the category

This category encompasses all pages undergoing significant changes. Template [[Template:Revision]] adds pages to this category.
[[Category:Pages needing attention]]

File:Solifuge.png

2026-08-29T09:40:19Z

Remu: Remu uploaded a new version of File:Solifuge.png

== Summary ==
Image of Solifuge

File:Solifuge.png

2026-08-29T09:36:00Z

Remu: Remu uploaded a new version of File:Solifuge.png

== Summary ==
Image of Solifuge

File:Solifuge.png

2026-08-29T09:34:43Z

Remu: Remu uploaded a new version of File:Solifuge.png

== Summary ==
Image of Solifuge

File:25p8.png

2026-08-29T07:08:44Z

Remu:

File:Annelid.png

2026-08-29T07:00:57Z

Remu: Remu reverted File:Annelid.png to an old version

== Summary ==
My nickname for this massive new ship that got constructed

File:Annelid.png

2026-08-29T07:00:36Z

Remu: Remu uploaded a new version of File:Annelid.png

== Summary ==
My nickname for this massive new ship that got constructed

Category:Puffers with period 1048576

2026-08-29T05:29:19Z

Remu: Created page with "{{PufferPeriod|p=1048576}}"

Category:Puffers with speed 256c/1048576

2026-08-29T05:28:54Z

Remu: Catagory

User talk:Resu

2026-08-28T13:41:58Z

Remu:

Please put new topics at the bottom of the page, sign your comments ( <nowiki>~~~~</nowiki> ), and do not edit other people's comments.
=Talk page taking ↓=

Bubblegum

2026-08-28T05:45:50Z

Remu: There are at least three red links within the wiki pointing to their name, excluding links created by that person (LifeWiki:Notability#Person_notability_guideline)

{{DISPLAYTITLE:bubblegum}}
{{stub}}
{{Person|name=bubblegum}}
'''bubblegum''' is a [[Life enthusiast]], known for their work with [[glider syntheses]]. They have also organized the [[OCA Discovery of the Year]] competitions for 2019 and 2020.
{{PatternsFoundBy}}

Superstring

2026-08-28T05:30:21Z

Remu: /* References */ Add references

{{Glossary}}
A '''superstring''' is an infinite orthogonal row of [[cell]]s stabilized on one side so that it moves at the [[speed of light]], often leaving debris behind. The first examples were found in {{year|1971}} by Edward Fitzgerald and [[Robert Wainwright]]. Superstrings were studied extensively by [[Peter Rott]] between {{year|1992}} and {{year|1994}}, and he found examples with many different [[period]]s. In August {{year|1998}}, [[Stephen Silver]] proved that odd-period superstrings are impossible.<ref name="post145357" />

==Waveguides==
Sometimes a finite section of a superstring can be made to run between two tracks, known as '''waveguide'''. This gives a [[fuse]] that can be made as wide as desired. The [[waveguide 1|first example]] was found by Tony Smithurst and uses [[tub]]s. Shortly after seeing this example, in March 1997 [[Peter Rott]] found another superstring track consisting of [[boat]]s. At present these are the only two orthogonal waveguides known. Both are destroyed by the superstring as it moves along — it would be interesting to find one that remains intact.

{{EmbedViewer
|pname = waveguide1
|viewerconfig = #C [[ WIDTH 850 HEIGHT 400 THUMBSIZE 2 ZOOM 14 ]]
|position = center
|caption = [[Waveguide 1]], a superstring running between two rows of tubs
|style = width:300px;
}}

The first example of a diagonal waveguide was discovered by [[Resu]] in July 2025 and uses extended [[barge]]s. It is also the third waveguide known overall. At present this is the only known [[clean]] diagonal waveguide.<ref name="post213949" />

{{EmbedViewer
|rle = #N Diagonal waveguide
x = 77, y = 93, rule = B3/S23
75bo$74bobo$73bobo$72bobo$71bobo$70bobo$69bobo$68bobo$67bobo$66bobo$65b
obo$64bobo$63bobo$62bobo$61bobo$60bobo$59bobo$58bobo$57bobo$56bobo$55b
obo$54bobo$53bobo$52bobo$51bobo$50bobo$49bobo$48bobo$47bobo$46bobo26b
o$45bobo26bobo$44bobo26bobo$43bobo26bobo$42bobo26bobo$41bobo26bobo$40b
obo26bobo$39bobo26bobo$38bobo26bobo$37bobo26bobo$36bobo26bobo$35bobo26b
obo$34bo2bo25bobo$33bo3bo24bobo$31b3o3bo23bobo$29b3obo3bo22bobo$29bob
obo3bo21bobo$29b2o2bo3bo20bobo$33bo3bo19bobo$33bo3bo18bobo$27bo5bo3bo
17bobo$33bo3bo16bobo$25bo7bo3bo15bobo$29bo3bo3bo14bobo$25bo3bo3bo3bo13b
obo$25bo3bo3bo3bo12bobo$25bo3bo3bo3bo11bobo$25bo3bo3bo3bo10bobo$25bo3b
o3bo3bo9bobo$25bo3bo3bo3bo8bobo$25bo3bo3bo3bo7bobo$25bo3bo3bo3bo6bobo
$25bo3bo3bo3bo5bobo$25bo3bo3bo3bo4bobo$25bo3bo3bo3bo3bobo$14bo10bo3bo
3bo3bo2bobo$12bo2bo2bobo4bo3bo3bo3bobobo$11bo3bobo7bo3bo3bo3b2obo$10b
o6bo3bo3bo3bo3bo3bobo$13bo3bo3bo3bo3bo3bo3b2o$9bo3bo4bobo4bo3bo3bo3bo
$9bo3bo5bo6bobo5bobo$6b2obo3bo13bo6b3o$5bo2b2o3bo$5bobo2bobo$6bo3bo20b
3o$9b2o19bo2bo$8b4o16bo4bo$8b5o2b3o10b3o$11bobob3o13bo$12b3obo7bo4b2o
$12bo6b2o2bobo3bo$12bo5bo6bob2o$19bo4bo2b2o$25bobo$20bo3b2o$20bo5bo$21b
o3bo$b2o2b2o$b2o3b2o6bo7b2o$bo3b2o7bo$o2b4o7bobo$bo2b2o$2b3o!
|viewerconfig = #C [[ WIDTH 850 HEIGHT 900 THUMBSIZE 2 ZOOM 9 ]]
|position = center
|caption = A superstring running between two diagonal rows of extended [[barge]]s
|style = width:300px;
}}

==See also==
* [[Titanic toroidal traveler]]
* [[Waveguide 1]]

==External links==
*[https://w.wiki/EcYg Waveguide] on Wikipedia
*[https://w.wiki/6DMS Superstring theory] on Wikipedia

==References==
<references>
<ref name="post145357">{{LinkForumThread
|format = ref
|p = 145357
|title = Re: Suggested LifeWiki edits
|author = Matthias Merzenich
|date = May 10, 2022
}}</ref>
<ref name="post213949">{{LinkForumThread
|format = ref
|p = 213949
|title = Re: Thread for basic questions
|author = Resu
|date = July 01, 2025
}}</ref>
</references>

==External links==
* {{LinkLexicon|lex_s.htm#superstring}}

__NOTOC__

Superstring

2026-08-28T05:29:38Z

Remu: /* Waveguides */

{{Glossary}}
A '''superstring''' is an infinite orthogonal row of [[cell]]s stabilized on one side so that it moves at the [[speed of light]], often leaving debris behind. The first examples were found in {{year|1971}} by Edward Fitzgerald and [[Robert Wainwright]]. Superstrings were studied extensively by [[Peter Rott]] between {{year|1992}} and {{year|1994}}, and he found examples with many different [[period]]s. In August {{year|1998}}, [[Stephen Silver]] proved that odd-period superstrings are impossible.<ref name="post145357" />

==Waveguides==
Sometimes a finite section of a superstring can be made to run between two tracks, known as '''waveguide'''. This gives a [[fuse]] that can be made as wide as desired. The [[waveguide 1|first example]] was found by Tony Smithurst and uses [[tub]]s. Shortly after seeing this example, in March 1997 [[Peter Rott]] found another superstring track consisting of [[boat]]s. At present these are the only two orthogonal waveguides known. Both are destroyed by the superstring as it moves along — it would be interesting to find one that remains intact.

{{EmbedViewer
|pname = waveguide1
|viewerconfig = #C [[ WIDTH 850 HEIGHT 400 THUMBSIZE 2 ZOOM 14 ]]
|position = center
|caption = [[Waveguide 1]], a superstring running between two rows of tubs
|style = width:300px;
}}

The first example of a diagonal waveguide was discovered by [[Resu]] in July 2025 and uses extended [[barge]]s. It is also the third waveguide known overall. At present this is the only known [[clean]] diagonal waveguide.<ref name="post213949" />

{{EmbedViewer
|rle = #N Diagonal waveguide
x = 77, y = 93, rule = B3/S23
75bo$74bobo$73bobo$72bobo$71bobo$70bobo$69bobo$68bobo$67bobo$66bobo$65b
obo$64bobo$63bobo$62bobo$61bobo$60bobo$59bobo$58bobo$57bobo$56bobo$55b
obo$54bobo$53bobo$52bobo$51bobo$50bobo$49bobo$48bobo$47bobo$46bobo26b
o$45bobo26bobo$44bobo26bobo$43bobo26bobo$42bobo26bobo$41bobo26bobo$40b
obo26bobo$39bobo26bobo$38bobo26bobo$37bobo26bobo$36bobo26bobo$35bobo26b
obo$34bo2bo25bobo$33bo3bo24bobo$31b3o3bo23bobo$29b3obo3bo22bobo$29bob
obo3bo21bobo$29b2o2bo3bo20bobo$33bo3bo19bobo$33bo3bo18bobo$27bo5bo3bo
17bobo$33bo3bo16bobo$25bo7bo3bo15bobo$29bo3bo3bo14bobo$25bo3bo3bo3bo13b
obo$25bo3bo3bo3bo12bobo$25bo3bo3bo3bo11bobo$25bo3bo3bo3bo10bobo$25bo3b
o3bo3bo9bobo$25bo3bo3bo3bo8bobo$25bo3bo3bo3bo7bobo$25bo3bo3bo3bo6bobo
$25bo3bo3bo3bo5bobo$25bo3bo3bo3bo4bobo$25bo3bo3bo3bo3bobo$14bo10bo3bo
3bo3bo2bobo$12bo2bo2bobo4bo3bo3bo3bobobo$11bo3bobo7bo3bo3bo3b2obo$10b
o6bo3bo3bo3bo3bo3bobo$13bo3bo3bo3bo3bo3bo3b2o$9bo3bo4bobo4bo3bo3bo3bo
$9bo3bo5bo6bobo5bobo$6b2obo3bo13bo6b3o$5bo2b2o3bo$5bobo2bobo$6bo3bo20b
3o$9b2o19bo2bo$8b4o16bo4bo$8b5o2b3o10b3o$11bobob3o13bo$12b3obo7bo4b2o
$12bo6b2o2bobo3bo$12bo5bo6bob2o$19bo4bo2b2o$25bobo$20bo3b2o$20bo5bo$21b
o3bo$b2o2b2o$b2o3b2o6bo7b2o$bo3b2o7bo$o2b4o7bobo$bo2b2o$2b3o!
|viewerconfig = #C [[ WIDTH 850 HEIGHT 900 THUMBSIZE 2 ZOOM 9 ]]
|position = center
|caption = A superstring running between two diagonal rows of extended [[barge]]s
|style = width:300px;
}}

==See also==
* [[Titanic toroidal traveler]]
* [[Waveguide 1]]

==External links==
*[https://w.wiki/EcYg Waveguide] on Wikipedia
*[https://w.wiki/6DMS Superstring theory] on Wikipedia

==References==
<references>
<ref name="post145357">{{LinkForumThread
|format = ref
|p = 145357
|title = Re: Suggested LifeWiki edits
|author = Matthias Merzenich
|date = May 10, 2022
}}</ref>
</references>

==External links==
* {{LinkLexicon|lex_s.htm#superstring}}

__NOTOC__

Superstring

2026-08-27T17:40:43Z

Remu: /* Waveguides */ It seems like LifeWiki:Notability#General notability guideline allows it. Please revert if it doesn't.

{{Glossary}}
A '''superstring''' is an infinite orthogonal row of [[cell]]s stabilized on one side so that it moves at the [[speed of light]], often leaving debris behind. The first examples were found in {{year|1971}} by Edward Fitzgerald and [[Robert Wainwright]]. Superstrings were studied extensively by [[Peter Rott]] between {{year|1992}} and {{year|1994}}, and he found examples with many different [[period]]s. In August {{year|1998}}, [[Stephen Silver]] proved that odd-period superstrings are impossible.<ref name="post145357" />

==Waveguides==
Sometimes a finite section of a superstring can be made to run between two tracks, known as '''waveguide'''. This gives a [[fuse]] that can be made as wide as desired. The [[waveguide 1|first example]] was found by Tony Smithurst and uses [[tub]]s. Shortly after seeing this example, in March 1997 [[Peter Rott]] found another superstring track consisting of [[boat]]s. At present these are the only two orthogonal waveguides known. Both are destroyed by the superstring as it moves along — it would be interesting to find one that remains intact.

{{EmbedViewer
|pname = waveguide1
|viewerconfig = #C [[ WIDTH 850 HEIGHT 400 THUMBSIZE 2 ZOOM 14 ]]
|position = center
|caption = [[Waveguide 1]], a superstring running between two rows of tubs
|style = width:300px;
}}

The first example of a diagonal waveguide was discovered by [[Resu]] in July 2025 and uses extended [[barge]]s. At present this is the only known
[[clean]] diagonal waveguide.

{{EmbedViewer
|rle = #N Diagonal waveguide
x = 77, y = 93, rule = B3/S23
75bo$74bobo$73bobo$72bobo$71bobo$70bobo$69bobo$68bobo$67bobo$66bobo$65b
obo$64bobo$63bobo$62bobo$61bobo$60bobo$59bobo$58bobo$57bobo$56bobo$55b
obo$54bobo$53bobo$52bobo$51bobo$50bobo$49bobo$48bobo$47bobo$46bobo26b
o$45bobo26bobo$44bobo26bobo$43bobo26bobo$42bobo26bobo$41bobo26bobo$40b
obo26bobo$39bobo26bobo$38bobo26bobo$37bobo26bobo$36bobo26bobo$35bobo26b
obo$34bo2bo25bobo$33bo3bo24bobo$31b3o3bo23bobo$29b3obo3bo22bobo$29bob
obo3bo21bobo$29b2o2bo3bo20bobo$33bo3bo19bobo$33bo3bo18bobo$27bo5bo3bo
17bobo$33bo3bo16bobo$25bo7bo3bo15bobo$29bo3bo3bo14bobo$25bo3bo3bo3bo13b
obo$25bo3bo3bo3bo12bobo$25bo3bo3bo3bo11bobo$25bo3bo3bo3bo10bobo$25bo3b
o3bo3bo9bobo$25bo3bo3bo3bo8bobo$25bo3bo3bo3bo7bobo$25bo3bo3bo3bo6bobo
$25bo3bo3bo3bo5bobo$25bo3bo3bo3bo4bobo$25bo3bo3bo3bo3bobo$14bo10bo3bo
3bo3bo2bobo$12bo2bo2bobo4bo3bo3bo3bobobo$11bo3bobo7bo3bo3bo3b2obo$10b
o6bo3bo3bo3bo3bo3bobo$13bo3bo3bo3bo3bo3bo3b2o$9bo3bo4bobo4bo3bo3bo3bo
$9bo3bo5bo6bobo5bobo$6b2obo3bo13bo6b3o$5bo2b2o3bo$5bobo2bobo$6bo3bo20b
3o$9b2o19bo2bo$8b4o16bo4bo$8b5o2b3o10b3o$11bobob3o13bo$12b3obo7bo4b2o
$12bo6b2o2bobo3bo$12bo5bo6bob2o$19bo4bo2b2o$25bobo$20bo3b2o$20bo5bo$21b
o3bo$b2o2b2o$b2o3b2o6bo7b2o$bo3b2o7bo$o2b4o7bobo$bo2b2o$2b3o!
|viewerconfig = #C [[ WIDTH 850 HEIGHT 900 THUMBSIZE 2 ZOOM 9 ]]
|position = center
|caption = A superstring running between two diagonal rows of extended [[barge]]s
|style = width:300px;
}}

==See also==
* [[Titanic toroidal traveler]]
* [[Waveguide 1]]

==External links==
*[https://w.wiki/EcYg Waveguide] on Wikipedia
*[https://w.wiki/6DMS Superstring theory] on Wikipedia

==References==
<references>
<ref name="post145357">{{LinkForumThread
|format = ref
|p = 145357
|title = Re: Suggested LifeWiki edits
|author = Matthias Merzenich
|date = May 10, 2022
}}</ref>
</references>

==External links==
* {{LinkLexicon|lex_s.htm#superstring}}

__NOTOC__

Superstring

2026-08-27T17:09:35Z

Remu: /* Waveguides */ Correction.

User:Resu/Sandbox

2026-08-27T17:02:03Z

Remu:

Edit[[Glider]]

<DynamicPageList>
namespace = Template
category = Templates
</DynamicPageList>

User:Resu

2026-06-26T13:41:59Z

Remu:

__NOEDITSECTION__
Welcome to my user-page. You might notice this is created by [[User:Remu]], but that's because I lost the password to this account.
==To-do==
*Continue with making the page for Heights
==Things==
*My first post is older then by join date by 16 years and 4 months.
==Links==
* [[User:Resu/Sandbox]]

User:Resu/Sandbox

2026-05-21T15:26:25Z

Remu: Test edit.

Edit[[Glider]]]]

<DynamicPageList>
namespace = Template
category = Templates
</DynamicPageList>

User:Resu/Dictionary

2026-03-02T17:23:32Z

Remu: Created page with "A dictionary of random, perhaps non-notable terms that I have seen in the CA community. ==E== ===Eonian=== Term used to describe objects that are more common than eater 1 and have a lower construction cost. [https://conwaylife.com/forums/viewtopic.php?f=2&t=7042#p220668]"

A dictionary of random, perhaps non-notable terms that I have seen in the CA community.
==E==
===Eonian===
Term used to describe objects that are more common than [[eater 1]] and have a lower construction cost. [https://conwaylife.com/forums/viewtopic.php?f=2&t=7042#p220668]

Superstring

2026-03-01T07:41:08Z

Remu: /* Waveguides */

User:Resu/Sandbox

2026-03-01T07:37:27Z

Remu:

<DynamicPageList>
namespace = Template
category = Templates
</DynamicPageList>

User:Resu/Sandbox

2026-03-01T07:36:42Z

Remu:

<DynamicPageList>
namespace = Template
category = Toplevel category
</DynamicPageList>

OCA:Travelling Ts

2026-02-28T19:53:06Z

Remu: /* References */ Removed errors

{{Rule
|name = Travelling Ts
|imgname = Travelling Ts
|char = Chaotic
|b = 3
|s = 23-a5
|reversal = B01245a78/S01234678
}}
'''Travelling Ts''' is an [[isotropic non-totalistic cellular automaton]] named in reference to its most common spaceship, known as the T. In May {{year|2025}}, a forum thread was started for this rule,<ref name="post212502" /> which ended up getting over 800 posts by the end of the year.
{| style="margin:auto; width:300px"
|{{EmbedViewer
|rle = x = 28, y = 5, rule = B3/S23-a5
7bo8bo8b2o$3o5b2o4b2ob2o4b2ob2o$bo5b2o6bobo6bob2o2$24bo!
|position = center
|caption = From left to right: the four most common [[spaceships]] are the Travelling T, the [[Glider]], the Climbing C, and the Flying F.
}}
|}

== Patterns ==
=== Oscillators ===
Travelling Ts is similar to [[Life]], meaning that some oscillators from Life work in this rule. In addition, all of the margolus oscillators from {{rl|2×2}} work in this rule<ref name="post212541" />, which enables the construction of oscillators with all even periods<ref name="post128639" /><ref name="post214235" />. Below is a collection of oscillators in this rule:
{| style="margin:auto; width:300px"
|{{EmbedViewer
|rle = x = 75, y = 73, rule = B3/S23-a5
68b2o$41b2o26bo$26b2o11bo2bo23b2o$2b3o3b4o14bo13bo$24bobo13bo23bobo$24b
2o13bo$21b2o18bo20bobo$20bobo17bo$2b4o14bo19bo19bobo$2b4o13b2o17bo2bo
18bo$38b2o18bo$58b2o3$24bo35bo$23bobo33bobo$22bobobo32b2o$20b3o33bo$5b
2o13bobo15bo17b7o$2bobob2o15b3o12b2o2b2o19bo$2bo5bo31bobo15b2o3bo$3b2o
bobo30bobo15bo6b2o$4b2o17b3o12bobo16b2o7bo$20bobo14b2o2b2o23bo2bo$20b
3o19bo23bobobo$22bobobo37bobob2o$23bobo36bobobo$24bo37b2o2bo$66b2o4$16b
2o40bo9bo$15bo2bo40b2o5b2o$4bo10bobo40bob2o3b2obo$3b2o11bo3b2o2b2o21b
o10bo9bo$4bo16bo2bobo15b2obo2bo5bob2obo2bobo2bob2obo$bo16bo2bo3b2o15b
o6bo5bo2bo3bobo3bo2bo$3o15b3o14b2o7b2ob2o6b2o4bo3bo4b2o$7b3o24bobo6b2o
11bo3bo5bo3bo$8bo15b3o7b2o22b2o7b2o$5bo12b2o3bo2bo10b2o2bo2bo$5b2o11b
obo2bo13bo5b2o13b2o7b2o$5bo13b2o2b2o3bo11bo15bo3bo5bo3bo$27bobo9bobo13b
2o4bo3bo4b2o$26bo2bo10bobo12bo2bo3bobo3bo2bo$27b2o12bo12bob2obo2bobo2b
ob2obo$58bo9bo$58bob2o3b2obo$59b2o5b2o$58bo9bo4$67b2o$3b2o26b2o34bo$3b
2o10bo16bo33b3o$3b2o9bo2bob2o9bo34bo$3b2o8bo6bo7bobo16b2o17b5o$3b2o9b
2ob2o24bobo2bo13bo7bo$3b2o13b2o6bobo13b2ob3o13bob5o2bobobo$3b2o36b2o19b
o5bobob3o$3b2o13bo2bo2bobo17b2o16bo2bo2bobobo$3b2o13b2o21b2obobo12bo2b
obobobo2bo$3b2o18b2o17bo2bo12bobobo2bo2bo$3b2o16bo18bobo13b3obobo5bo$
3b2o16b2o17b2o14bobobo2b5obo$60bo7bo$60b5o$65bo$62b3o$63bo$62b2o!
|position = center
|caption = Smallest known oscillators from periods 2 to 17.
}}
|}
An [[edgy]] [[dot]] (or in some cases, [[thumb]], [[duoplet]], or [[domino]]) can [[reflect]] a T 90 degrees, enabling the creation of T [[Glider loop oscillator|loop]]s.
The [[sparkers]] below enable the creation of T loops of [[period]]s 3n+30, 5n+30, 6n+30, 7n+35, 8n+32, 10n+30, 11n+33, 20n+40, 24n+48, 25n+50, and 32n+32:
{{EmbedViewer
|rle = x = 283, y = 29, rule = B3/S23-a5
2o18b2o20b2o18b2o33b2o26b2o29b2o39b2o20b2o20b2o22b2o$3o17b3o19b3o17b3o
32b3o25b3o28b3o38b3o19b3o19b3o21b3o$2o18b2o20b2o18b2o33b2o26b2o29b2o39b
2o20b2o20b2o22b2o$280bo$6bo19bo19bo3bo18bo34bo26bo35bo11bo22bo22bo3bo
44bo5b2o$b2o3bo3b2o14bo15b2obo5bob2o14bo7bo26bo26bo35bo11bo22bo22bo3b
o17b2o6b2o17bo6b2o$bo9bo8b2o9b2o9bo11bo19bo2b3o3b2o80bobo11bobo19b3o18b
o9bo14b2o6b2o20b2o$2b4ob4o10bo2bobobo2bo11b3obobob3o13bo3bo2b3o3bo2bo
14bo3b5o3bo21b2o31bo2bo9bo2bo19bo22bo5bo14b2o3bo2bo3b2o17b2o$5bobo12b
o3b5o3bo14bobo15b3obob3o4b2obobo13bobobobobobobobo16bobob2obo27b2obob
o9bobob2o17bobo18bo2bo3bo2bo12b2o2bo4bo2b2o13b2o6bo$4bo3bo11b4o5b4o10b
4o3b4o10bo4bo4b4obobo2b2o10bo2bobobobobobo2bo15b2o3bo2bo28bo15bo18bob
o19b3o5b3o15bo6bo17b2o5bo$4b2ob2o14bo5bo13bo9bo10b2ob2ob3o8b2obo12b2o
2bo2bo2bo2b2o16b2o5bobo23b3o4bob2o3b2obo4b3o15b2ob2o41bo8bo17bo$20b4o
2bo2b4o11bo7bo15bobo3b2obobo4bo16bob3obo28bobo25b2obo2b2o3b2o2bob2o20b
o$19bo2bo2bobo2bo2bo9b2o7b2o14bobobo4b2obobob2o11b4ob2ob2ob4o25bobo32b
obo$19b2o5bo5b2o33b2obob4o6bobo11bo4bo5bo4bo25bobo26b2ob3ob3ob2o67bo8b
o$71bo4b6o2bo11b3ob2o5b2ob3o54b2obobobobob2o68bo6bo$73bo2bo2bo4b2o7b2o
bo2b2o7b2o2bob2o23b3o27b3o3b3o44b3o5b3o12b2o2bo4bo2b2o$72b2o22b3ob2o5b
2ob3o27b2o80bo2bo3bo2bo12b2o3bo2bo3b2o$96bo4bo5bo4bo30b2o24b3o3b3o46b
o5bo16b2o6b2o$97b4ob2ob2ob4o28bo26b2obobobobob2o42bo9bo14b2o6b2o$101b
ob3obo30b2o27b2ob3ob3ob2o45bo3bo$97b2o2bo2bo2bo2b2o26b3o31bobo50bo3bo
$96bo2bobobobobobo2bo51b2obo2b2o3b2o2bob2o$97bobobobobobobobo49b3o4bo
b2o3b2obo4b3o$98bo3b5o3bo54bo15bo$163b2obobo9bobob2o$104bo60bo2bo9bo2b
o$104bo60bobo11bobo$167bo11bo$167bo11bo!
|viewerconfig = #C [[ HEIGHT 400 WIDTH 3600 THUMBSIZE 3 ]]
|position = center
|caption = Sparkers reflecting Ts
|style = width:300px;
}}
So far, only one stable [[circuitry]] component has been found, and it has only been used to create a p248 oscillator<ref name="post223837" />. In Travelling Ts, [[reaction]]s tend to not grow as large as they would in standard [[Life]], resulting in fewer opportunities for [[catalyst]] placements.
=== Spaceships ===
In addition to the [[spaceship]]s displayed above, there are lots of other, larger spaceships. A collection of useful spaceships is displayed below:
{| style="margin:auto; width:300px"
|{{EmbedViewer
|viewerconfig = #C [[ HEIGHT 360 WIDTH 2400 THUMBSIZE 2 ]]
|rle = x = 206, y = 33, rule = B3/S23-a5
184bo$182b4o$179b3obo2b2obo$182bo3bob3o$179bo2bo5bo2b2o$4bo5bo12b3o7b
3o15bo2bo15bo4bo18b5o14bo8b3o12bo11bo29bo4bobob3o2bo$3b3o3b3o11bo2bo2b
o2bo2bo14b2o2b2o13b3o2b3o16b2obob2o12b3o7b2ob2o9bobo9bobo6bo6bo12b2ob
o3b2ob3o4bo$2b2o2bobo2b2o8bobo2bobobobo2bobo10b3ob2ob3o10bo2b4o2bo16b
o3bo13b3o9b2ob2o6bo2bo9bo2bo4b2o6b2o13bo2bobo5b3o2bo$2bo3bobo3bo8bo2b
obobobobobo2bo9b2obo4bob2o31b3obo3bob3o21b3obo7b2o11b2o5bobo4bobo9bo2b
o4bo4b3o2bo2bo$3ob2o3b2ob3o5b2ob2o3bobo3b2ob2o52b2obo3bob2o25bo10bo9b
o6b3ob4ob3o8b2o5bob3obobo3bo2bo$4b3ob3o10bo2bob3ob3obo2bo10bo8bo14b2o
20bobobo11b3ob3o5b2o14b3o5b3o6b12o28bo2bo$6bobo12b2o3bobobobo3b2o8bob
o8bobo10b2o2b2o13b2obo3bo3bob2o7b2ob2o9bo12b2o7b2o5b2ob2o4b2ob2o16bob
o6bo2bo2bo$4bo5bo10bo4bobobobo4bo6bo2bo10bo2bo7b2o4b2o18b3o15bo11bo12b
4o3b4o5b2o2bo4bo2b2o16bob2o6bo2bo2bo$4bo5bo11b2o4bobo4b2o7b3o12b3o8bo
b2obo13b5o5b5o33b3o2bobo2b3o5bobobo2bobobo14b4o2b2o9bo2bo$22b2o4bobo4b
2o34bo2bo14b2o3b5o3b2o20b3o10b3o2bobo2b3o5b2o3b2o3b2o14bobo2bo11bo2bo
$27bo3bo35b2ob6ob2o11bob4ob4obo21bobo14b2ob2o12b6o16b3o13b3obobo$23b2o
3bobo3b2o35bo2bo20bobo26b3o28bo3bo2bo3bo11b3o16bob2ob2o$23bo2b3ob3o2b
o51b2o2b2o7b2o2b2o35b5o10b3o4b3o26bo3b2ob4o$23b2o3bobo3b2o55b2o7b2o39b
obobo10b2obo2bob2o14bo11b2o5bo3bo$26b2obob2o56b3o9b3o19bobobo63bob2ob
o3bob2o$21b5o7b5o86bobo69bo2bo4bo$20bo5b7o5bo83bob3obo63b3obob3o3b2o$
19b2ob4o7b4ob2o50bo11bo19b2o3b2o62b2o4bo5b2o$19b2obo3bo5bo3bob2o51b2o
7b2o20b2o3b2o61bobo3bobo2b2obo$24b3o2bo2b3o57bo7bo22b2ob2o62b3o10bo$27b
5o60bo7bo86bob2o8bo3bo$25b3o3b3o59bo2bo2bo25bo63bo8bob2obo$25bo7bo58b
2obobob2o24bo63bo7bobo$93bobobobo99bo$92b2obobob2o95bo$92bo2bobo2bo95b
2o$93bo5bo$93bo5bo!
|caption = Many of the spaceships that were discovered in this rule. Note that this collection does not include larger spaceships.
}}
|}
The [[c/2|4c/8]] spaceship has occurred [[seminatural|seminaturally]]<ref>{{cata|object/xq8_8mlp34043plm8zx9y19/b3s23-a5|xq8_8mlp34043plm8zx9y19 on Catagolue}}</ref>, and the [[3c/8]] has a useful [[spark]] that can be used in technology to create [[rake]]s and [[puffer]]s.
===Infinite growth===
A variety of [[linear growth]] patterns are known, including [[gun]]s<ref name="post214903" />, [[puffer]]s<ref name="post212651" />, [[rake]]s<ref name="post212502"/>, [[tubstretcher]]s<ref name="post213858" />, [[growing spaceship]]s<ref name="post213591" />, and a [[slide gun]].
Below is a gallery of small linear growth patterns:
{{gallery item|{{EmbedViewer
|rle = x = 51, y = 85, rule = B3/S23-a5
21b2o5b2o$20b2ob2ob2ob2o$20b2obo3bob2o2$22b2o3b2o$22b3ob3o$24b3o$25bo
2$23b2ob2o$20bo4bo4bo$18b2ob4ob4ob2o$19bo3b5o3bo$21bobobobobo$23bo3bo$
22b2o3b2o$22b3ob3o$17b3o2bobobobo2b3o$18bobobo5bobobo$20bo2bo3bo2bo$
18bobo9bobo$19b2o9b2o$20b2o7b2o$19b3obo3bob3o$22bo5bo$18b2o2bo5bo2b2o$
18bob2obo3bob2obo$20b2o7b2o7$18bo13bo$16bo2bo11bo2bo$16b2o15b2o$17bobo
11bobo$17bobo11bobo8$13bo23bo$12bo25bo$12b3o21b3o10$9bo31bo$8bo33bo$8b
3o29b3o10$5bo39bo$4bo41bo$4b3o37b3o10$bo47bo$o49bo$3o45b3o!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 6 GPS 6 TRACKLOOP 16 0/16 -8/16 Y -20 ]]
|position = left
|caption = A p16 c/2 backrake, found using ikpx2
|style = width:300px;
|apgcode = yl16_1_10_b7e8c7057c5317a06464c1ad917f8c64/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 66, y = 65, rule = B3/S23-a5
3o$o$bo3bo$3b2ob2o2$5bo2bo2$7b2o$8bo$9bo$10b2o$10b2o$11b2o$13b2o$14b3o
3$15b3o$19b2o$19bo2bo$19bobo$22bo$21bo$21b2obo$25bo$24b2o$25b2o$27bob
o$28bo$28bo$31b4o$31bob3o$31b2obobo$31b6o$32bob3obo$33b7o2bo$36bo2b3o
bo$35b2obob2obo$36b2o3bo$37b2o3bo2bo$37b3o2b3ob2o$36bo3b2ob2ob2o$37b2o
2b2ob2o$41b3o3b2obo$40bo2bo3bobobo$41b2o7bobo$41b2ob2o2bobo$44bo2b2ob
o$45bo4bo$44bob4o$45bo6bo$46bo4bobo$52bobo$53bobo$54bobo$55bobo$56bob
o$57bobo$58bobo$59bobo$60bobo$61bobo$62bobo$63bobo$64bo!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 7 GPS 12 X -5 Y -5 TRACKLOOP 4 -1/4 -1/4 ]]
|position = right
|caption = A c/4d [[tubstretcher]], found using ikpx2
|style = width:300px;
|apgcode = yl4_iforgottheapgcode/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 126, y = 126, rule = B3/S23-a5
5b3o$5bo$6bo$8b2o$9bo$2o5b2ob2o$obo$o4bo4bobo$3bobo6b2o$3b2o6bo$5bobo
2bob2obo$5bo3bo2b2obo$7b2ob2o3b2o$8bob2o4b3o$16b2o$10b3o3bo$12b4o$13b
2o4b2o$13bo4bob2o$17bobobo2bobo$17b2o6b2o$18b2o2bobo$21bo3b2obo$24b2o$
19bobobobo2bo$20bob3o4bo$19b2obo4b2o2bo$26bo2bobo$22bobobobo2bo$25bobo
3bo$31bo$26b8o$31bo2b2o$31bo2b5o$32b2obob3o$32b3ob2o3bo$33bobobo5bo$
33b4o2b2obo$33b2o4b4o$34bo2b2obobob2o$37b3o4b2o$35bo2bo7bo$37b3o4bobo$
36bo6b2obo$39b2ob2o2bo2bo$39b2o5b5o$41b5obo$45b5obo$45bobob3o$44b2ob2o
bobo$45bo2b2ob3ob2o$47b2obo4bobo$49b2o$50bo5bo$56b2o$50b2o3b2o3bo$50bo
2b4ob2o$51bo2bo7bo$56bob2obo$56bobo2bo$55bo4b4o$58b3ob2o$57bo2b2o2bo$
60b2o2bo3b2o$62b2ob4ob2o$64b2o2bob2o$64bobob4o$64bo2b2o$63b7o$63bo2bob
o2bo$64b3o5bo$64b3o2bo2bo$70b2o2b2o$74b2obo$72b2o4bo$72b2o3bo$77b4o$
73bob2o4bo$74bobo3bobo$76bo4bobo$76bobo3bobo$77bobo3bobo$78bobo3bobo$
79bobo3bobo$80bobo3bobo$81bobo3bobo$82bobo3bobo$83bobo3bobo$84bobo3bob
o$85bobo3bobo$86bobo3bobo$87bobo3bobo$88bobo3bobo$89bobo3bobo$90bobo3b
obo$91bobo3bobo$92bobo3bobo$93bobo3bobo$94bobo3bobo$95bobo3bobo$96bobo
3bobo$97bobo3bobo$98bobo3bobo$99bobo3bobo$100bobo3bobo$101bobo3bobo$
102bobo3bobo$103bobo3bobo$104bobo3bobo$105bobo3bobo$106bobo3bobo$107bo
bo3bobo$108bobo3bobo$109bobo3bobo$110bobo3bobo$111bobo3bobo$112bobo3bo
bo$113bobo3bobo$114bobo3bobo$115bobo3bobo$116bobo3bobo$117bobo3bobo$
118bobo3b2o$119bobo$120bobo$121b2o!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 4 GPS 12 TRACKLOOP 4 -1/4 -1/4 Y -20 ]]
|position = right
|caption = A c/4d double [[tubstretcher]], found using ikpx2
|style = width:300px;
|apgcode = yl4_1_4_239f5fb6f41059f03c0b7bf04ded49a8/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 112, y = 139, rule = B3/S23-a5
12b3o$11bo2bob2o$10b2o3bob3o$10bob2ob2obobo$10b2ob2obob2o$8b2obo4b2o$
7b4obo2bo$6b2o5bob2o2b2o$5b2o12bo$5b2o2b2o2b5o3bobo$2b3obo2bo5bo2b3o2b
o$b2ob2o9bo2b2o4b2o$o2bo2bo13bo$o2b2o2bobo9bob2o$2o2bo4bo11b2o$2b2o2b
2ob3o6b2o$bob3obobo9bobo$b2o2bo3bo6b2o3bo$2b3o5b2o3b3obo$2bobo3bo4b2o
6bo$3bo3b4o3bo2b2o2bo2b3o$8bobo2b3obo3bob4o$8bo5b3o5bob3o$8bo2bo3bo2b
2o2b2o4b2o$10bo4bob3o3bo2bob2o$8bob2o3bo6bo2b2o$7bobobo6b2obo4bob2o$6b
o4b2obo2b3obo2bo2b2o$6bo2bo2bobo3b2o4b5o$6bo2bobo2b4ob2obo3bo$12bo3bo
5bo2bo$13b6o3bo5bo$13b4o7b2obo$14bo2bob3obob2obobo$14b4o3b2ob3o$14b2o
5bobo4b2o$18b2obob2o2b3o$16bo4bob6o$15bo4b3o2b2obo$14b2o5b3obo2bo2bo$
15b2o3b6o2b2o2bo$15b2ob2obo2bobo4bo2bo$15b2o2b2obo4b2o4bo$15bob4o4bobo
5bobo$21bobo3bo4b4o$16bobo3bo9bo4b2o$15b3o2bo2bo7bo2bo$15bo2bob2o10bob
o$16bo4b2o7bo2bo$18bo2bo3b3o2b2o$21bo3bo7bobo$19b3o3bo2bobo2bobo$23b2o
2b2ob2o3bo$21bobo3b2ob3obobo$21b2o6b2obo$22bo5b4o2b2o$22bo5bo3b2ob3o$
23b2o3b2o3b2obo$27bo4bobo3b2o$25bobo10bo$28b2o3bo$32b2obob2o$31b3ob3o$
37bo$38b3o$38b2o$41b2o$40b3o$45bo$44b2o$43b3o$42b2o$41b3o2bo$45bobo$
46bobo$47bobo$48bobo$49bobo$50bobo$51bobo$52bobo$53bobo$54bobo$55bobo$
56bobo$57bobo$58bobo$59bobo$60bobo$61bobo$62bobo$63bobo$64bobo$65bobo$
66bobo$67bobo$68bobo$69bobo$70bobo$71bobo$72bobo$73bobo$74bobo$75bobo$
76bobo$77bobo$78bobo$79bobo$80bobo$81bobo$82bobo$83bobo$84bobo$85bobo$
86bobo$87bobo$88bobo$89bobo$90bobo$91bobo$92bobo$93bobo$94bobo$95bobo$
96bobo$97bobo$98bobo$99bobo$100bobo$101bobo$102bobo$103bobo$104bobo$
105bobo$106bobo$107bobo$108bobo$109bobo$110b2o!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 4 GPS 12 TRACKLOOP 3 -1/3 -1/3 Y -20 ]]
|position = left
|caption = A c/3d [[tubstretcher]], found using ikpx2 (a reduction is known; see the next gallery item)
|style = width:300px;
|apgcode = yl3_1_2_68230f624fb7a1ce0993de5602efde3f/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 96, y = 39, rule = B3/S23-a5
14b2obo$10b3obob6o$9bo2bobo3b3obo$8b2o6bo4b2o$7b2obob2o5bo2b2o$6b2o4b
obo4b2o4bo$5b2o3b2ob2o4b2obob2o$4b2o5b2o$3b2o4bo2bo7bo2bo$2b2o4bob2o3b
2o4bobobo$bo2bobo2bo2b3o2bo7b2obo$bo4b2obo4b2o8b6o$b2ob2ob2obo2b3o9b2o
bo2bo$4bobo3bob3o10b2o2bobo$3o2b2o3b5ob2o7b2o3bo$o8bob2o5bo10bo$bobo5b
o4bo3bo9bo2b2o$2o8bo3bo3b3o7bo4bo$b2o12b5o8bo3bo$b2ob3o10b2o$b2o2b2ob
o8bo3b2ob2o$bobo5bo10bob2obo$2b3obo13b2obo2b2o$4bo3b2o11b2obo2b6obo$6b
o4bo8bo2bo2bob2o4b2obo$5b2o2b6o5b2o3b4o3bo2bo2bo2b2o$10b5o7bob2o2b2ob
obobo4b2obo$11bo10b2obobobo3bo5bobo$10b3o3b3o4b4o4bob2o3b3obo$11bobob
o7b2ob2o4b2obo2bo2bobo5b3o14b3o14b2o3b2o$14bo8bo8bobo4bo2bobo4bo6b3o7b
o15b2o3b2o$12b2o2bo6bo2bobo6bo7bobo4bo5bo10bo10b3o2bobo3bo$16bobo4bob
o3b2o5b2o6bobo5bo5bob3o6bo7bo2bo3bo$17bo8b4o6b2o7bobo3b3obo4b3ob2o2b4o
b6o5bo5b2o$23b2o3bobo15bobo2bobo4bo3b4o2bobob2ob3o8bo2b2o2bo$24b3o2bo
bo15bobo4b3o6bobobo3bo4bo2b2o5b2o4b4o$32b2o14b3o3b2ob2o7bo4b3ob3ob2o7b
o2bo3bo$24bo7b2o15b2o16b2o2bo4b2ob2o11bo$25bo!
|viewerconfig = #C [[ AUTOSTART AUTOFIT WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 5 GPS 12 TRACKLOOP 200 -1/3 -1/3 Y -20 ]]
|position = left
|caption = A growing spaceship with frontend speed c/3d and backend speed c/4d, made by attaching a c/4d tubeater to a c/3d tubstretcher
|style = width:300px;
|apgcode =
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 49, y = 86, rule = B3/S23-a5
20b2o5b2o$19b2ob2ob2ob2o$19b2obo3bob2o2$21b2o3b2o$21b3ob3o$23b3o$24bo
2$22b2ob2o$19bo4bo4bo$17b2ob4ob4ob2o$18bo3b5o3bo$20bobobobobo$22bo3bo$
21b2o3b2o$21b3ob3o$16b3o2bobobobo2b3o$17bobobo5bobobo$19bo2bo3bo2bo$
17bobo9bobo$18b3o7b3o$20bo7bo$19b2o7b2o$19bo9bo3$17b3o9b3o$17b2o2bo5bo
2b2o$15b2ob2obo5bob2ob2o$2b3o3b3o4b2o2bo9bo2b2o4b3o3b3o$bo2bo3bo2bo4b
2o13b2o4bo2bo3bo2bo$bo3bobo3bo5bobo9bobo5bo3bobo3bo$3bobobobo8bo11bo8b
obobobo$2o2bo3bo2b2o23b2o2bo3bo2b2o$obob2ob2obobo23bobob2ob2obobo$3bob
obobo29bobobobo$4bo3bo31bo3bo$15b2o15b2o$14bo2bo13bo2bo$14bo2bo13bo2bo
$15b2o15b2o5$15b2o15b2o$14bo2bo13bo2bo$14bo2bo13bo2bo$15b2o15b2o5$15b
2o15b2o$14bo2bo13bo2bo$14bo2bo13bo2bo$15b2o15b2o2$20b3o$20bo$21bo$15b
2o15b2o$14bo2bo13bo2bo$14bo2bo13bo2bo$15b2o15b2o2$20b3o$20bo$21bo$15b
2o$14bo2bo$14bo2bo$15b2o2$20b3o$20bo$21bo$15b2o$14bo2bo$14bo2bo$15b2o
2$20b3o$20bo$21bo!
|viewerconfig = #C [[ WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 4 ]]
|position = right
|caption = A growing spaceship, with a frontend speed of c/2 and a backend speed of 32c/114.
|style = width:300px;
|apgcode = yl3_1_2_68230f624fb7a1ce0993de5602efde3f/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 106, y = 106, rule = B3/S23-a5
5b3o$5bo$6bo$8b2o$9bo$2o5b2ob2o$obo$o4bo4bobo$3bobo6b2o$3b2o6bo$5bobo
2bob2obo$5bo3bo2b2obo$7b2ob2o3b2o$8bob2o4b3o$16b2o$10b3o3bo$12b4o$13b
2o4b2o$13bo4bob2o$17bobobo2bobo$17b2o6b2o$18b2o2bobo$21bo3b2obo$24b2o$
19bobobobo2bo$20bob3o4bo$19b2obo4b2o2bo$26bo2bobo$22bobobobo2bo$25bobo
3bo$31bo$26b8o$31bo2b2o$31bo2b5o$32b2obob3o$32b3ob2o3bo$33bobobo5bo$
33b4o2b2obo$33b2o4b4o$34bo2b2obobob2o$37b3o4b2o$35bo2bo7bo$37b3o4bobo$
36bo6b2obo$39b2ob2o2bo$39b2o5bob2obo$41b5o2bo3bo$51b3o$45b2o3bo$45bo5b
o$48bo3bo$45bobobo3bo$46b2o2bo3bo$47bo3bo3bo$52bo3bo$53bo3bo$54bo3bo$
55bo3bo$56bo3bo$57bo3bo$58bo3bo$59bo3bo$60bo3bo$61bo3bo$62bo3bo$63bo3b
o$64bo3bo$65bo3bo$66bo3bo$67bo3bo$68bo3bo$69bo3bo$70bo3bo$71bo3bo$72bo
3bo$73bo3bo$74bo3bo$75bo3bo$76bo3bo$77bo3bo$78bo3bo$79bo3bo$80bo3bo$
81bo3bo$82bo3bo$83bo3bo$84bo3bo$85bo3bo$86bo3bo$87bo3bo$88bo3bo$89bo3b
o$90bo3bo$91bo3bo$92bo3bo$93bo3bo$94bo3bo$95bo3bo$96bo3bo$97bo3bo$98bo
3bo$99bo3bobo$100bo3b2o$101bo$102bo$101b2o!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 4 GPS 12 TRACKLOOP 4 -1/4 -1/4 Y -20 ]]
|position = right
|caption = A c/4d double [[linestretcher]], found using ikpx2
|style = width:300px;
|apgcode = yl4_1_2_d900f3c2a36dc20a1459e0d2d1c66e69/b3s23-a5
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 111, y = 100, rule = B3/S23-a5
2b30o$2b30o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o
30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$
2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b
2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o
30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$
2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b
2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o30b2o$2o
30b2o$2o30b2o$2o30b2o$2b30o$2b30o12$20b2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o
6b2o6b2o6b2o6b2o$20b3o5b3o5b3o5b3o5b3o5b3o5b3o5b3o5b3o5b3o5b3o5b3o$20b
2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o6b2o4$14bo3bo$10b2obob3obob
2o$9bobobo2bo2bobobo$9b2o2bobobobo2b2o$13bobobobo$9b4obobobob4o$8bo4bo
5bo4bo$9bob2o7b2obo$9b2o2bo5bo2b2o$9bob2o7b2obo$8bo4bo5bo4bo$9b4obobob
ob4o$13bobobobo$9b2o2bobobobo2b2o$9bobobo2bo2bobobo$10b2obob3obob2o$
14bo3bo!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 4 GPS 4 LOOP 32 X -20 ]]
|position = right
|caption = A p32 T gun
|style = width:300px;
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 178, y = 219, rule = B3/S23-a5
67bo$67bo$63bobo2bo$61b3o2b2o$63b3obo$66bo4$47b3o$46b2ob2o$46b3ob2o$
50b2o$48b3o$49b2o$48bo14$26b2o$2b3o21b3o$b2ob2o20b2o$b2o2bo$2bobo$2b3o
2$54b4o$54bo2b2o$53bo5bo$54b2o2bo$54bo3$4b2o$3bobo$3bo$2b2o2$38b3o$37b
2ob2o$36b2ob3o$36b2o$37b3o$37b2o$39bo32bob2o$72b2obo$66b2o5bobobo$65b
3o5bo3bo$66b2o6b3o$75b2o5$42b2o$42bo$43b3o$45bo19$31bo$31b3o$34bo$33b
2o3$53bo$2b2o48bobo$2o49b2ob2ob2o$2ob3o45bobo3b3o$bo3bo43b2o6b3o$2b3o
44b3o7bo$4b2o44b2o3b4o$55bo$55bo$38b2o$37b5o$36b2o2b2o$36bobobobo$36b
3o$37b2o5$70bobo$71b2o$71bo$22bo$20b2ob2o$20bo2bo$20bo$20b2o$21bobo$
22bo$71b2o$69b2ob2o$69bo3bo$69b5o$71b2o15$26b2o$24b5o$16b2o6b2o2b2o$
15b3o5bobobobo$16b2o9b3o$27b2o$11bo$11bo$10b3o$12bo$10bobo$11bobo$8b2o
b2o$10bo63$175bobo$176b2o$176bo!
|viewerconfig = #C [[ AUTOSTART WIDTH 500 HEIGHT 500 THUMBSIZE 2 ZOOM 3 GPS 42 LOOP 420 X -20 ]]
|position = right
|caption = A p420 glider gun, which uses a G+T->2G+2T [[over-unity reaction]].
|style = width:300px;
}}}}
==Universality==
On December 20th, 2025, [[PK22]] posted a [[Rule 110]] [[metacell]], proving the rule [[Universal computer|Turing-complete]].<ref name="post222945" />
== References ==
<references>
<ref name="post212502">{{LinkForumThread
|format = ref
|p = 212502
|title = Travelling Ts(b3s23-a5)
|author = ThePlayzr
|date = May 29, 2025
}}</ref>
<ref name="post212541">{{LinkForumThread
|format = ref
|p = 212541
|title = Re: Travelling Ts(b3s23-a5)
|author = hotcrystal0
|date = May 30, 2025
}}</ref>
<ref name="post128639">{{LinkForumThread
|format = ref
|p = 128639
|title = Re: Miscellaneous Discoveries in Other Cellular Automata
|author = creeperman7002
|date = April 21, 2021
}}</ref>
<ref name="post223837">{{LinkForumThread
|format = ref
|p = 223837
|title = Re: Travelling Ts(b3s23-a5)
|author = R2INT
|date = January 10, 2026
}}</ref>
<ref name="post214235">{{LinkForumThread
|format = ref
|p = 214235
|title = Re: Travelling Ts(b3s23-a5)
|author = PK22
|date = July 7, 2025
}}</ref>
<ref name="post214903">{{LinkForumThread
|format = ref
|p = 214903
|title = Re: Travelling Ts(b3s23-a5)
|author = Luka Okanishi
|date = July 21, 2025
}}</ref>
<ref name="post212651">{{LinkForumThread
|format = ref
|p = 212651
|title = Re: Travelling Ts(b3s23-a5)
|author = PK22
|date = May 31, 2025
}}</ref>
<ref name="post213858">{{LinkForumThread
|format = ref
|p = 213858
|title = Re: Travelling Ts(b3s23-a5)
|author = PK22
|date = June 29, 2025
}}</ref>
<ref name="post213591">{{LinkForumThread
|format = ref
|p = 213591
|title = Re: Travelling Ts(b3s23-a5)
|author = PK22
|date = June 22, 2025
}}</ref>
<ref name="post222945">{{LinkForumThread
|format = ref
|p = 222945
|title = Re: Travelling Ts(b3s23-a5)
|author = PK22
|date = December 20, 2025
}}</ref>
</references>

== External Links ==
* {{LinkForumThread|f=11|t=6970|title=Travelling Ts (B3/S23-a5) and its variants}}
* {{LinkCatagolueRule|b3s23-a5}}

[[Category:Isotropic non-totalistic rules]]
[[Category:2×2 block rules]]
__NOTOC__

Eater/Snippet

2026-02-28T17:09:33Z

Remu: Edited to match the Eater page.

<noinclude>{{Snippet}}</noinclude>[[Image:Eater5_small.gif|right]]An '''eater''' is any [[catalyst]] that can delete residual [[pattern]]s produced by certain reactions. The term may also sometimes specifically refer to [[eater 1]], a very common and well-known eater. The [[block]] was the first known eater, being found to be capable of eating [[beehive]]s from a [[queen bee]], allowing the construction of the [[queen bee shuttle]]. The animation to the right shows an [[eater 5]] feasting on an incoming stream of [[glider]]s.

Eaters are extremely important, as they help stabilize and control debris created by complex reactions, allowing for the manipulation of the useful parts of those reactions. [[Stable reflector]]s in particular heavily rely on a variety of eaters to work.

User:Resu/Sandbox

2026-02-28T17:05:33Z

Remu: Created page with "<DynamicPageList> category = Everything else </DynamicPageList>"

<DynamicPageList>
category = Everything else
</DynamicPageList>

User:Resu/Draft/Test

2026-02-28T17:04:53Z

Remu: Remu moved page User:Resu/Sandbox to User:Resu/Draft/Test without leaving a redirect

{{SpaceshipNavbox}}
{{CategoryTools}}
This category contains all notable {{lc:{{SUBPAGENAME}}}}.

User:Resu

2026-02-28T17:03:22Z

Remu:

__NOEDITSECTION__
Welcome to my user-page. You might notice this is created by [[User:Remu]], but that's because I lost the password to this account.
==To-do==
*Need to do the Spaceships eaters collection's c/3 section
*Continue with making the page for Heights
==Things==
*My first post is older then by join date by 16 years and 4 months.
==Links==
* [[User:Resu/Sandbox]]

P5 phase shifter

2026-02-28T16:58:37Z

Remu: Zoom

{{Oscillator
|name = p5 phase shifter
|pname = 31p5phaseshifter
|c = 31
|bx = 9
|by = 12
|p = 5
|m = 5
|h = 2.8
|v = 0.20
|sv = 0.20
|symmetry = n
|discoverer = Dean Hickerson
|discoveryear = 1997
|rulemin = B3/S23
|rulemax = B378/S023678
|isorulemin = B3aejnr/S2-n3-aci
|isorulemax = B2in34eq5cij6-ek78/S01c234-jntz5678
|rle = true
|plaintext = true
|apgcode = xp5_8k4ra1vgc453z116871
|viewerconfig = #C [[ THUMBSIZE 2 ZOOM 30 GPS 3 HEIGHT 500 ]]
}}
'''p5 phase shifter''' is a {{period|5}} [[oscillator]] that can provide a domino spark; when it does, it gets shifted three generations forward, allowing for 5n+3 [[LCM]]-type oscillators. The oscillator itself was discovered by [[Dean Hickerson]] in {{year|1997}}, being among the first oscillators found with [[dr]].<ref>{{cite web|title=New Billiard Tables|url=https://conwaylife.com/ref/DRH/nbt.html|author=Dean Hickerson}}</ref> The phase-shifting reaction was found by [[Matthias Merzenich]] on August 13, {{year|2021}}.<ref name="post134826" />

The [[snake]] motif can be replaced with a [[fishhook]] for added [[clearance]] at the cost of increasing the population from 31 cells to 32.

==Rephasing reactions==

Here are examples of [[SKOP]]s which use the phase-shift capabilities
{{gallery top}}
{{gallery item|{{EmbedViewer
|rle = x = 16, y = 12, rule = B3/S23
3b2o4b2o$4bo6bo$2bo5bo$2b2o4bo3bo$8bo3b4o$3obo5bo$o2bobo$bobob3o$2ob2o
3bo$2bo3b2o$2bob2o$3bobo!
|position = center
|caption = p18 phase shifter with [[Caterer]], tied for SKOP18 with [[p18 bi-block hassler]]. Shows the aircraft carrier version of the oscillator
|style = width:300px;
|viewerconfig = [[ THUMBSIZE 2 ZOOM 20 GPS 16 HEIGHT 500 ]]
|apgcode = xp18_wc93xs112ogggzbdhe96kk801zw1213
}}}}
{{gallery item|{{EmbedViewer
|pname = 64p63
|caption = 64P63, uses [[32P21]]
|style = width:300px;
|position = center
|viewerconfig = [[ THUMBSIZE 2 ZOOM 18 GPS 16 HEIGHT 500 ]]
|apgcode = xp63_8o6pajkk8z32q921w4a4z321w66y1eh9mzy83443wgw33zyd121
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 22, y = 19, rule = B3/S23
3bobo$2bob2o$2bo3b2o$2ob2o3bo$bobob3o$o2bobo15bo$3obo14b3o$18bo$2b2o14b2o$2bo
$obo5b3o$2o7bo$16bo$15b3o2$6b2o$7bo$4b3o$4bo!
|caption = 54P108, uses [[22P36]]
|style = width:300px;
|position = center
|viewerconfig = [[ THUMBSIZE 2 ZOOM 18 GPS 16 HEIGHT 500 ]]
|apgcode = xp108_8o6pajkk8z32q921y8ca23z321y1131y08c8zy0c453
}}}}
{{gallery item|{{EmbedViewer
|rle = x = 30, y = 24, rule = B3/S23
3bobo$2bob2o$2bo3b2o$2ob2o3bo$bobob3o$o2bobo$3obo2$2b2o$2bo$obo7b2o$2o
7bo2bo$12bo$9b3o2$7b2o12b2o$7b2o12b2o4$4b2o18bo$2o4b4o10b2o2b2ob3o$2o
2b2ob3o10b2o4b4o$4bo19b2o!
|caption = 79P288, uses [[Two blockers hassling R-pentomino]]
|style = width:300px;
|position = center
|viewerconfig = [[ THUMBSIZE 2 ZOOM 14 GPS 16 HEIGHT 500 ]]
|apgcode = xp288_8o6pajkk8z32q921z321y2a996zy333y833z66wd52666y666wba4666
}}}}
{{gallery item|{{EmbedViewer
|position = center
|rle = x = 59, y = 45, rule = B3/S23
23b2o16b2o$23b2o7b2o7b2o$32b2o5$7b2o48b2o$7b2o48b2o2$3bo$2bobo20bo14b
o$2bobo20b3o10b3o$2o2b2o$bobo2b2o18b3o8b3o$o2b2obobob2o$b2o3bob2obo$3b
2obo$3bob2o$15b2o32b2o$15b2o32b2o4$15b2o32b2o$15b2o32b2o5$26b3o8b3o2$
25b3o10b3o$25bo14bo3$7b2o48b2o$7b2o48b2o5$32b2o$23b2o7b2o7b2o$23b2o16b
2o!
|viewerconfig = #C [[ GPS 16 ZOOM 8 THUMBSIZE 2 HEIGHT 500 ]]
|caption = 115P398, uses [[84P199]] and the snake form of the oscillator.<ref name="post192909" />
|style = width:300px;
|apgcode = xp398_yc33y366y333zccyzy5cczye6kkgy4gkk6ydgg8eh6o8zy4ggysggx31230f85d221zy4hhyshhzy411ys11zyec551y4155cz66yzy566zycooy3ccy3oo
}}}}
{{gallery bottom}}

Other period-multiplying rephasings which aren't SKOP include the following<ref name="post219837"/>:
{| class="sortable wikitable" style="margin-left:auto;margin-right:auto;"
!Original Oscillator
!Original Period
!Original Offset from 5n
!Multiplied period
|-
![[Mold]]
|4
|5n+4
|{{LinkCatagolue|xp8_wdbx2dg94ozbdhf97kk8011zw1213|patternname=8|style=raw}}
|-
![[34P7]]
|7
|5n+2
|{{LinkCatagolue|xp28_660g0fmz0cqb0t53zw6fx66wg8gzy0o8go0sabchmazy53213|patternname=28|style=raw}}
|-
![[Lumps of muck hasslers#p37|p37 lumps of muck hassler]]
|37
|5n+2
|{{LinkCatagolue|xp148_8o6pajkk8z32qa06w8k4ozw23y21woyj39u0o4cz03h4u0uiy0166yc3r4y02303146zw11w346ylo48gzyv11|patternname=148|style=raw}}
|-
![[Twin bees shuttle]]
|46
|5n+1
|{{LinkCatagolue|xp138_66ybb9ey366zooybk4sybbdzyd11y78kk69fhdbzyt3121|patternname=138|style=raw}}
|-
![[p47 pre-pulsar shuttle]]
|47
|5n+2
|{{LinkCatagolue|xp188_y9252y1252zy1ggweydewggzx4701xggw4r4x4r4wggx1074zx4s0gx11w4r4x4r4w11xg0s4z31e8011weydew11zbdhe96kk8y08k8y18k8zw1213|patternname=188|style=raw}}
|}
Note rephasings do not work to multiply periods for most standard sparking oscillators with periods of the form 5n+3 as the period will ultimately line up with the spark every time. Likewise, the rephasing will not work outright for oscillators with periods which are multiples of 5 because the oscillator will no longer be in phase after the first interaction.

== References ==
<references>
<ref name="post134826">{{LinkForumThread
|format = ref
|date = August 13, 2021
|p = 134826
|title = Re: Oscillator Discussion Thread
|author = Matthias Merzenich
}}</ref>
<ref name="post192909">{{LinkForumThread
|format = ref
|date = August 25, 2024
|p = 192909
|title = Re: Smallest Known Oscillators to p106 (and Beyond)
|author = Period1GliderGun
}}</ref>
<ref name="post219837">{{LinkForumThread
|format = ref
|title = Re: Suggested LifeWiki edits
|p = 219837
|author = WhiteHawk
|date = October 17, 2025
}}</ref>
</references>

== External links ==
* {{LinkCatagolue|xp5_8k4ra1vgc453z116871|patternname=with fishhook}}
* {{LinkCatagolue|xp5_8k4ra1vgc48cz116871|patternname=with snake}}
* {{LinkCatagolue|xp5_8k4ra1vgc4goz116871|patternname=with aircraft carrier}}

[[Category:Oscillators]]
[[Category:Oscillators with period 5]]
[[Category:Oscillators with mod 5]]

LWSS-to-G

2026-02-28T16:38:06Z

Remu:

{{disambig|name=LWSS-to-G}}
* [[45-degree LWSS-to-G]]
* [[135-degree LWSS-to-G]]

==External links==
{{LinkLexicon|lex_l.htm#lwsstog|name=LWSS-to-G}}

Z

2026-02-28T16:36:06Z

Remu:

{{disambig|name=Z}}
* [[Z-tetromino]]
* [[Z-pentomino]]
* [[Z-hexomino]]
* [[Z quad-loaf]]

Category:Spaceships with unsimplified speed 55c/309174

2026-02-28T09:44:08Z

Remu: Created page with "{{RawSpeedCategory|s=55c/309174}}"

Category:Spaceships with unsimplified speed 2c/28409785756

2026-02-28T09:43:49Z

Remu: Created page with "{{RawSpeedCategory|s=2c/28409785756}}"

Category:Spaceships with speed c/14204892878

2026-02-28T09:43:12Z

Remu: Created page with "{{SpeedCategory|s=c/14204892878}}"

Category:Spaceships with speed 55c/309174

2026-02-28T09:42:57Z

Remu: Created page with "{{SpeedCategory|s=55c/309174}}"

Category:Spaceships with period 28409785756

2026-02-28T09:41:56Z

Remu: Created page with "{{SpaceshipPeriod|p=28409785756}}"

User:Resu/Spaceships with mod 28409785756

2026-02-28T09:24:49Z

Remu: Created page with "{{User:Resu/Sandbox}}"

User:Resu/Draft/Test

2026-02-28T09:23:57Z

Remu:

{{SpaceshipNavbox}}
{{CategoryTools}}
This category contains all notable {{lc:{{SUBPAGENAME}}}}.

User:Resu/Draft/Test

2026-02-28T09:23:00Z

Remu: Template testing

{{SpaceshipNavbox}}
{{CategoryTools}}
This category contains all notable {{lc:{{PAGENAME}}}}.

Category:Spaceships with mod 28409785756

2026-02-28T09:18:27Z

Remu: Created page with "{{SpaceshipModCategory|m=28409785756}}"

Category:Spaceships with mod 154587

2026-02-28T09:18:05Z

Remu: Created page with "{{SpaceshipModCategory|m=154587}}"

Static symmetry

2026-02-28T09:15:56Z

Remu: /* Higher Dimensions */

A '''static symmetry'''<ref name="post138823" /> 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 non-totalistic cellular automaton|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 ==
[[File:Hickerson_and_Catagolue_symmetries.png|center|frame|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#Kinetic_symmetry_naming_system|kinetic symmetry]] article.

=== Rotational ===
<div class="mw-collapsible">
In Catagolue notation, rotational symmetries are prefixed with "C", referring to the cyclic groups.<ref>{{LinkWikipedia|Cyclic_group|name=Cyclic group|nobullet=nobullet}}</ref>
<div class="mw-collapsible-content">

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

<center>{{Sym-demo
|rle = x = 16, y = 16, rule = B/S01234V
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o!
|width = 160
|height = 160
|caption = C1 symmetry
}}</center>

==== 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$15b16o!
|width = 320
|height = 320
|caption = C2_1 symmetry}}
| {{Sym-demo
|rle = x = 31, y = 32, rule = B/S01234V
16o$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$15b16o!
|width = 320
|height = 320
|caption = C2_2 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 32, rule = B/S01234V
16o$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!
|width = 320
|height = 320
|caption = 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$31o!
|width = 320
|height = 320
|caption = C4_1 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 32, rule = B/S01234V
32o$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!
|width = 320
|height = 320
|caption = C4_4 symmetry}}
|}
</div></div>

C4 tends to produce record [[diehard]]s and [[megasized]] soups compared to other symmetries.

=== Reflectional ===
<div class="mw-collapsible">
In Catagolue notation, reflectional symmetries are prefixed with "D", referring to the dihedral groups.<ref>{{LinkWikipedia|Dihedral_group|name=Dihedral group|nobullet=nobullet}}</ref>
<div class="mw-collapsible-content">

==== 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$16o!
|width = 160
|height = 320
|caption = D2_+1 symmetry}}
| {{Sym-demo
|rle = x = 16, y = 32, rule = B/S01234V
16o$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!
|width = 160
|height = 320
|caption = D2_+2 symmetry}}
|}

* '''D2_x''' The line is diagonal. Example: [[Elevener]]

<center>{{Sym-demo
|rle = x = 16, y = 16, rule = B/S01234V
16o$o14bo$o14bo$o7bo6bo$o6bobo5bo$o7bo3bo2bo$o11bo2bo$o3bo7bo2bo$o2bob
o6bo2bo$o3bo6bo3bo$o9bo4bo$o8bo5bo$o4b4o6bo$o14bo$o14bo$16o!
|width = 160
|height = 160
|caption = D2_x symmetry}}</center>

==== 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$31o!
|width = 320
|height = 320
|caption = D4_+1 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 31, rule = B/S01234V
32o$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$32o!
|width = 320
|height = 320
|caption = D4_+2 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 32, rule = B/S01234V
32o$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!
|width = 320
|height = 320
|caption = 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$31o!
|width = 320
|height = 320
|caption = D4_x1 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 32, rule = B/S01234V
32o$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!
|width = 320
|height = 320
|caption = 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]]

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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$31o!
|width = 320
|height = 320
|caption = D8_1 symmetry}}
| {{Sym-demo
|rle = x = 32, y = 32, rule = B/S01234V
32o$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!
|width = 320
|height = 320
|caption = D8_4 symmetry}}
|}

</div></div>

== On a hexagonal or triangular grid ==
[[File:Hexagonal symmetries.png|center|frame|Overview of hexagonal symmetries. Blue arrows correspond to non-normal subgroups.]]

[[Hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by planar<ref>{{LinkMathworld|filename=DualTessellation.html|pagename=Dual tessellation}}</ref> or polytopic duality<ref>{{LinkWikipedia|Dual_polyhedron#Dual_polytopes_and_tessellations|name=Dual polyhedron|nobullet=nobullet}}</ref> - 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.<ref name="post66638" />

=== Rotational ===
<div class="mw-collapsible">
Click on "Expand" to the right to view a list of hexagonal/triangular rotational symmetries.
<div class="mw-collapsible-content">

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

<center>{{Sym-demo
|rle = 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!
|width = 300
|height = 300
|caption = C1 symmetry
}}</center>

==== 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.

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = C2_1 symmetry}}
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = 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)

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = C3_1 symmetry}}
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = C3_3 symmetry}}
|}

==== C6 ====
'''C6''': Symmetric under 60° rotation.

<center>{{Sym-demo
|rle = 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!
|width = 500
|height = 500
|caption = C6 symmetry}}</center>
</div></div>

=== Reflectional ===
<div class="mw-collapsible">
Click on "Expand" to the right to view a list of hexagonal/triangular reflectional symmetries.
<div class="mw-collapsible-content">

==== 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).

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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!
|width = 300
|height = 300
|caption = D2_x symmetry}}
| {{Sym-demo
|rle = 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!
|width = 300
|height = 300
|caption = 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.

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = D4_x1 symmetry}}
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = 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)

{| style="margin-left: auto; margin-right: auto"
| {{Sym-demo
|rle = 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!
|width = 400
|height = 400
|caption = D6_1 symmetry}}
| {{Sym-demo
|rle = 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$15b16o!
|width = 400
|height = 400
|caption = D6_1o symmetry}}
| {{Sym-demo
|rle = x = 47, y = 47, rule = B/S0123HT
16o15b16o$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!
|width = 400
|height = 400
|caption = D6_3 symmetry}}
|}

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

<center>{{Sym-demo
|rle = 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!
|width = 500
|height = 500
|caption = D12 symmetry}}</center>
</div></div>

== Higher Dimensions ==
The symmetries for the cubic grid are listed at [[Cubic grid symmetries]].

== References ==
<references>
<ref name="post138823">{{LinkForumThread
|format = ref
|title = Re: Help with symmetries
|p = 138823
|author = GUYTU6J
|date = December 13, 2021
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
</references>

== External links ==
* {{LinkLexicon|lex_s.htm#symmetric|name=Symmetric}}
* {{LinkForumThread|f=7|t=1898|title=Help with symmetries}}

[[Category:Everything else]]

LifeViewer/User Interface

2026-02-28T09:15:20Z

Remu:

[[LifeViewer]] has a set of mouse controls and keyboard commands.
== Mouse Controls ==
In Pan mode click and drag with the mouse to pan the view. The mouse wheel can be used to zoom in or out of the pattern. The zoom will be centered on the mouse pointer.

In Draw mode click and drag with the mouse to draw cells.

In Select mode click and drag with the mouse to make a selection.

=== Top Row ===
[[File:ui-top.png]]

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the menu functions.
<div class="mw-collapsible-content">

Going from left to right the controls perform the following functions:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Auto</td><td>Toggle</td><td>toggles AutoFit mode</td></tr><tr>
<td>Fit</td><td>Button</td><td>fits the pattern to the window</td></tr><tr>
<td>Draw</td><td>Option</td><td>switches to Draw mode for drawing cells</td></tr><tr>
<td>Select</td><td>Option</td><td>switches to Select mode for copy/paste</td></tr><tr>
<td>Pan</td><td>Option</td><td>switches to Pan mode for viewing patterns</td></tr><tr>
<td>Zoom</td><td>Slider</td><td>changes the camera zoom</td></tr><tr>
<td>START</td><td>Toggle</td><td>toggles AUTOSTART mode (if defined)</td></tr><tr>
<td>STOP</td><td>Toggle</td><td>toggles STOP mode (if defined)</td></tr><tr>
<td>WAYPT</td><td>Toggle</td><td>toggles Waypoint mode (if defined)</td></tr><tr>
<td>LOOP</td><td>Toggle</td><td>toggles LOOP mode (if defined)</td></tr><tr>
<td>Icon</td><td>Toggle</td><td>toggles icon display</td></tr><tr>
<td>Grid</td><td>Toggle</td><td>toggles grid line display</td></tr><tr>
<td>Help</td><td>Toggle</td><td>toggles Help display</td></tr><tr>
</table>
</div>
</div>

=== Bottom Row ===
[[File:ui-bottom.png]]

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the menu functions.
<div class="mw-collapsible-content">

Going from left to right the controls perform the following functions:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>T 0</td><td>Toggle</td><td>toggles generation statistics display</td></tr><tr>
<td>Undo</td><td>Button</td><td>undo the last edit</td></tr><tr>
<td>Redo</td><td>Button</td><td>redo the last undone action</td></tr><tr>
<td>1x</td><td>Button</td><td>sets playback speed to 1x</td></tr><tr>
<td>1x</td><td>Slider</td><td>sets playback speed</td></tr><tr>
<td>Reverse</td><td>Button</td><td>reverse playback direction</td></tr><tr>
<td>Reset</td><td>Option</td><td>reset to T=0 and if at T=0 also reset camera</td></tr><tr>
<td>Step Back</td><td>Option</td><td>step back step size generations</td></tr><tr>
<td>Step Forward</td><td>Option</td><td>step forward step size generations</td></tr><tr>
<td>Play</td><td>Option</td><td>start playback</td></tr><tr>
<td>Settings</td><td>Toggle</td><td>toggles the settings menu</td></tr><tr>
</table>
</div>
</div>

=== Draw Mode ===
[[File:draw-mode.png]]

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the menu functions.
<div class="mw-collapsible-content">

Going from left to right the controls perform the following functions:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Pick</td><td>Toggle</td><td>click on a cell to set drawing state</td></tr><tr>
<td>State Display</td><td>Toggle</td><td>toggle state display</td></tr><tr>
<td>Draw Pause</td><td>Toggle</td><td>toggle pause playback while drawing</td></tr><tr>
<td>Smart Draw</td><td>Toggle</td><td>toggle smart drawing</td></tr><tr>
<td>0-6</td><td>Slider</td><td>change the displayed states</td></tr><tr>
<td>State List</td><td>Option</td><td>select the current drawing state</td></tr><tr>
</table>
</div>
</div>

=== Select Mode ===
[[File:select-mode.png]]

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the menu functions.
<div class="mw-collapsible-content">

Going from left to right the controls perform the following functions:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>All</td><td>Button</td><td>select all cells</td></tr><tr>
<td>Auto Shrink</td><td>Toggle</td><td>auto shrink selection to pattern</td></tr><tr>
<td>Shrink</td><td>Button</td><td>shrink selection to pattern</td></tr><tr>
<td>Clipboards</td><td>Toggle</td><td>toggle clipboard libary to select active clipboard</td></tr><tr>
<td>Cut</td><td>Button</td><td>cut cells in selection to current clipboard</td></tr><tr>
<td>Copy</td><td>Button</td><td>copy cells in selection to current clipboard</td></tr><tr>
<td>Paste</td><td>Button</td><td>paste cells from current clipboard</td></tr><tr>
<td>Top Left</td><td>Slider</td><td>set paste location</td></tr><tr>
<td>AND/CPY/OR/XOR</td><td>Option</td><td>set paste mode</td></tr><tr>
</table>
</div>
</div>

=== Selection Menu ===
[[File:selection-menu.png]]

The Selection Menu appears when there is an active Selection in Select Mode.

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the menu functions.
<div class="mw-collapsible-content">

Going from top to bottom and left to right the controls perform the following functions:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Cancel</td><td>Button</td><td>cancel selection</td></tr><tr>
<td>Left</td><td>Button</td><td>nudge selection left 1 cell</td></tr><tr>
<td>Right</td><td>Button</td><td>nudge selection right 1 cell</td></tr><tr>
<td>Up</td><td>Button</td><td>nudge selection up 1 cell</td></tr><tr>
<td>Down</td><td>Button</td><td>nudge selection down 1 cell</td></tr><tr>
<td>50%</td><td>Slider</td><td>set random fill percentage</td></tr><tr>
<td>Random</td><td>Button</td><td>random fill selection using all states</td></tr><tr>
<td>Random 2</td><td>Button</td><td>random fill selection 2 states</td></tr><tr>
<td>Invert</td><td>Button</td><td>invert cells in selection</td></tr><tr>
<td>Clear History/Super</td><td>Button</td><td>clear [R]History/[R]Super cells</td></tr><tr>
<td>Flip X</td><td>Button</td><td>flip selection horizontally</td></tr><tr>
<td>Flip Y</td><td>Button</td><td>flip selection vertically</td></tr><tr>
<td>Rotate CW</td><td>Button</td><td>rotate selection clockwise</td></tr><tr>
<td>Rotate CCW</td><td>Button</td><td>rotate selection counter-clockwise</td></tr><tr>
<td>Clear Inside</td><td>Button</td><td>clear cells inside selection</td></tr><tr>
<td>Clear Outside</td><td>Button</td><td>clear cells outside selection</td></tr><tr>
</table>
</div>
</div>

=== Settings Menu ===
[[File:settings.png]]

The Settings Menu can be opened by clicking on the Settings button. Note the Settings button will not function if the Viewer window is less than 480 pixels high.
The menu contains four sliders and a set of topic buttons. Topics can contain settings and actions.

==== Sliders ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the slider functions.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Angle</td><td>Slider</td><td>set camera angle (square grid only)</td></tr><tr>
<td>Tilt</td><td>Slider</td><td>set camera tilt (square grid only)</td></tr><tr>
<td>Layers</td><td>Slider</td><td>set number of cell layers</td></tr><tr>
<td>Depth</td><td>Slider</td><td>set depth between each cell layer</td></tr><tr>
</table>
The Layers setting creates a pseudo-3D effect where cells are drawn in higher layers the longer they live, and appear to fall through lower layers the longer it has been since they died.
</div>
</div>

==== Pattern ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the Pattern options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Change Rule</td><td>Button</td><td>prompts for a new Rule and then switches to it</td></tr><tr>
<td>New Pattern</td><td>Button</td><td>creates a new blank Pattern with the current Rule</td></tr><tr>
<td>Load Pattern</td><td>Button</td><td>loads the last saved Pattern</td></tr><tr>
<td>Save Pattern</td><td>Button</td><td>saves the current Pattern to the text box</td></tr><tr>
<td>Rand All</td><td>Button</td><td>creates a new random Pattern with a random Rule (based on the current rule family)</td></tr><tr>
<td>Rand Pattern</td><td>Button</td><td>creates a new random Pattern with the existing Rule</td></tr><tr>
<td>Identify</td><td>Button</td><td>identify whether the current Pattern is a still life, oscillator or spaceship</td></tr><tr>
<td>Last Identify</td><td>Button</td><td>show last Identify results</td></tr><tr>
<td>Save Image</td><td>Button</td><td>save an image of the current pattern</td></tr><tr>
<td>Save Graph</td><td>Button</td><td>save an image of the Population Graph</td></tr><tr>
</table>
</div>
</div>

==== Clipboard ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the Clipboard options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Open Clipboard</td><td>Button</td><td>opens the clipboard as a pattern</td></tr><tr>
<td>Copy Original</td><td>Button</td><td>copy original pattern</td></tr><tr>
<td>Copy Rule</td><td>Button</td><td>copies the current Rule definition to the clikpboard</td></tr><tr>
<td>Copy NHood</td><td>Button</td><td>copy CoordCA neighbourhood definition</td></tr><tr>
<td>Copy All</td><td>Button</td><td>copy the current selection with comments</td></tr><tr>
<td>Paste To Seln</td><td>Button</td><td>paste clipboard to selection and tile if selection larger than clipboard</td></tr>
<td>External Sync</td><td>Toggle</td><td>sync with external clipboard</td></tr><tr>
</table>
</div>
</div>

==== Display ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the Display options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Use Rectangles</td><td>Toggle</td><td>toggle use of rectangular cells for Hexagonal and Triangular grids</td></tr><tr>
<td>Cell Borders</td><td>Toggle</td><td>toggles drawing of cell borders</td></tr><tr>
<td>Major GridLines</td><td>Toggle</td><td>toggle drawing major GridLines</td></tr><tr>
<td>Starfield</td><td>Toggle</td><td>toggle drawing starfield in background</td></tr><tr>
<td>Annotations</td><td>Toggle</td><td>toggle annotation display (if specified)</td></tr><tr>
<td>Rainbow</td><td>Toggle</td><td>toggle using Rainbow cell colours (for 2-state patterns)</td></tr><tr>
<td>Auto GridLines</td><td>Toggle</td><td>toggle automatically switching GridLines on for Draw and Select modes, and off for Pan mode</td></tr><tr>
<td>Alt GridLines</td><td>Toggle</td><td>toggle alternating GridLines (for Margolus patterns)</td></tr><tr>
<td>Integer Zoom</td><td>Button</td><td>set Zoom to nearest integer</td></tr><tr>
<td>Render Quality</td><td>Toggle</td><td>toggles anti-aliased cell display</td></tr>
</table>
</div>
</div>

==== Playback ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the Playback options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Go To Gen</td><td>Button</td><td>prompts for a generation and then moves to that generation</td></tr><tr>
<td>AutoFit History</td><td>Toggle</td><td>toggle using history states for AutoFit bounding box</td></tr><tr>
<td>Throttle</td><td>Toggle</td><td>toggle playback throttling</td></tr><tr>
<td>Perf. Warning</td><td>Toggle</td><td>toggle performance warning display</td></tr><tr>
<td>Kill Gliders</td><td>Toggle</td><td>toggle killing escaping gliders</td></tr><tr>
<td>AutoHide UI</td><td>Toggle</td><td>toggle hiding UI during playback</td></tr><tr>
</table>
</div>
</div>

==== General ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the General options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Fit Selection</td><td>Button</td><td>fit selection to display</td></tr><tr>
<td>Center Pattern</td><td>Button</td><td>center pattern on display</td></tr><tr>
<td>Clear Drawing</td><td>Button</td><td>clear drawing state cells in pattern</td></tr><tr>
<td>Snap Angle</td><td>Button</td><td>snap camera angle to nearest 45 degrees</td></tr><tr>
<td>Save View</td><td>Button</td><td>save camera view</td></tr><tr>
<td>Restore View</td><td>Button</td><td>restore saved camera view</td></tr><tr>
</table>
</div>
</div>

==== Advanced ====
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to see the Advanced options.
<div class="mw-collapsible-content">

<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Frame Times</td><td>Toggle</td><td>toggles display of the frames per second and CPU load %</td></tr><tr>
<td>Timing Details</td><td>Toggle</td><td>toggles extended timing information display</td></tr><tr>
<td>Relative Gen</td><td>Toggle</td><td>toggles absolute or relative generation display (when #CXRLE Gen specified)</td></tr><tr>
<td>Reset All</td><td>Button</td><td>reset all Viewers</td></tr><tr>
<td>Stop All</td><td>Button</td><td>stop all Viewers</td></tr><tr>
<td>Stop Others</td><td>Button</td><td>stop other Viewers</td></tr><tr>
<td>Info Bar</td><td>Toggle</td><td>toggles the Info Bar display which shows camera location, angle, and bounding box velocity</td></tr>
</table>
</div>
</div>

==== Theme ====
Allows selection from a list of available colour Themes.

==== Graph ====
Toggles the [[#Population Graph|Population Graph]].

=== Population Graph ===
LifeViewer can show a real time auto-scaling population graph. It is toggled on or off with [[#Graph|Settings→Graph]]. At each generation it shows total population, births and deaths.
Note that the graph will only capture data for the first 2<sup>19</sup> (524288) generations.

{{LV:Viewer|x = 9, y = 9, rule = x = 7, y = 3, rule = B3/S23|
o3b3o$3o2bo$bo!
[[ ZOOM 4 GRAPH LOOP 1000 HEIGHT 400 AUTOSTART ]]}}

Graph controls:
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Name</td><td bgcolor=#c0c0c0>Type</td><td bgcolor=#c0c0c0>Function</td></tr><tr>
<td>Opacity</td><td>Slider</td><td>set the graph opacity %</td></tr><tr>
<td>Pop</td><td>Toggle</td><td>toggle drawing population</td></tr><tr>
<td>Bth</td><td>Toggle</td><td>toggle drawing births</td></tr><tr>
<td>Dth</td><td>Toggle</td><td>toggle drawing deaths</td></tr><tr>
<td>Lines</td><td>Toggle</td><td>toggle between drawing lines or points</td></tr><tr>
<td>X</td><td>Button</td><td>close the graph</td></tr><tr>
</table>

== Keyboard commands ==

A visual map of keyboard shortcuts is shown below:

[[File:Hotkeys.png|1024px]]

Below is a list of keyboard commands by category:

=== Playback controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Playback controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Enter</td><td>toggle play / pause</td></tr><tr>
<td>Space</td><td>pause / next generation</td></tr><tr>
<td>B</td><td>pause / previous generation</td></tr><tr>
<td>Backspace</td><td>pause / previous generation</td></tr><tr>
<td>Backslash</td><td>pause / next step</td></tr><tr>
<td>Shift Backslash</td><td>pause / previous step</td></tr><tr>
<td>Esc</td><td>pause if playing</td></tr><tr>
<td>R</td><td>reset to generation 0</td></tr><tr>
<td>Shift N</td><td>go to generation</td></tr><tr>
<td>-</td><td>decrease playback speed</td></tr><tr>
<td>+</td><td>increase playback speed</td></tr><tr>
<td>Shift -</td><td>minimum playback speed</td></tr><tr>
<td>Shift +</td><td>maximum playback speed speed</td></tr><tr>
<td>0</td><td>reset playback speed</td></tr><tr>
<td>Ctrl P</td><td>toggle pause playback while drawing</td></tr><tr>
<td>Shift F7</td><td>toggle fast lookup</td></tr><tr>
<td>Alt T</td><td>toggle throttling</td></tr><tr>
<td>W</td><td>toggle waypoint playback and loop</td></tr><tr>
<td>Shift P</td><td>toggle just loop</td></tr><tr>
<td>Alt O</td><td>toggle autostart</td></tr><tr>
<td>Alt P</td><td>toggle stop</td></tr><tr>
</table>
</div>
</div>

=== Multi-Viewer controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Multi-viewer controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Shift R</td><td>reset all LifeViewers to generation 0</td></tr><tr>
<td>Z</td><td>stop playback in all other LifeViewers</td></tr><tr>
<td>Shift Z</td><td>stop playback in all LifeViewers</td></tr>
</table>
</div>
</div>

=== Camera controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Camera controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>V</td><td>restore saved camera position</td></tr><tr>
<td>Shift V</td><td>save current camera position</td></tr><tr>
<td>F</td><td>fit pattern to display</td></tr><tr>
<td>Shift F</td><td>toggle autofit</td></tr><tr>
<td>Shift H</td><td>toggle autofit history mode</td></tr><tr>
<td>Ctrl F</td><td>fit selection to display</td></tr><tr>
<td>Ctrl M</td><td>center pattern on display</td></tr><tr>
<td>[</td><td>zoom out</td></tr><tr>
<td>]</td><td>zoom in</td></tr><tr>
<td>Shift [</td><td>halve zoom</td></tr><tr>
<td>Shift ]</td><td>double zoom</td></tr><tr>
<td>1</td><td>1x zoom</td></tr><tr>
<td>2</td><td>2x zoom</td></tr><tr>
<td>4</td><td>4x zoom</td></tr><tr>
<td>8</td><td>8x zoom</td></tr><tr>
<td>6</td><td>16x zoom</td></tr><tr>
<td>3</td><td>32x zoom</td></tr><tr>
<td>Shift 3</td><td>64x zoom</td></tr><tr>
<td>Shift 1</td><td>integer zoom</td></tr><tr>
<td>Shift 2</td><td>-2x zoom</td></tr><tr>
<td>Shift 4</td><td>-4x zoom</td></tr><tr>
<td>Shift 8</td><td>-8x zoom</td></tr><tr>
<td>Shift 6</td><td>-16x zoom</td></tr><tr>
<td>Left</td><td>pan left</td></tr><tr>
<td>Right</td><td>pan right</td></tr><tr>
<td>Up</td><td>pan up</td></tr><tr>
<td>Down</td><td>pan down</td></tr><tr>
<td>Shift Left</td><td>pan north west</td></tr><tr>
<td>Shift Right</td><td>pan south east</td></tr><tr>
<td>Shift Up</td><td>pan north east</td></tr><tr>
<td>Shift Down</td><td>pan south west</td></tr><tr>
<td><</td><td>rotate left</td></tr><tr>
<td>></td><td>rotate right</td></tr><tr>
<td>Shift <</td><td>rotate left 90 degrees</td></tr><tr>
<td>Shift ></td><td>rotate right 90 degrees</td></tr><tr>
<td>Alt <</td><td>rotate left 45 degrees</td></tr><tr>
<td>Alt ></td><td>rotate right 45 degrees</td></tr><tr>
<td>5</td><td>reset angle</td></tr>
<td>Alt /</td><td>snap angle to nearest 45 degrees</td></tr>
<td>'</td><td>tilt down</td></tr><tr>
<td>/</td><td>tilt up</td></tr><tr>
<td>@</td><td>reset tilt</td></tr><tr>
</table>
</div>
</div>

=== View controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view View controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Q</td><td>increase number of layers</td></tr><tr>
<td>A</td><td>decrease number of layers</td></tr><tr>
<td>P</td><td>increase layer depth</td></tr><tr>
<td>L</td><td>decrease layer depth</td></tr><tr>
<td>C</td><td>next colour theme</td></tr><tr>
<td>Shift C</td><td>previous colour theme</td></tr><tr>
<td>Alt C</td><td>default theme</td></tr>
</table>
</div>
</div>

=== Display controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Display controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>G</td><td>toggle generation statistics</td></tr><tr>
<td>Shift G</td><td>toggle generation display mode</td></tr><tr>
<td>Y</td><td>toggle population graph</td></tr><tr>
<td>Shift Y</td><td>toggle graph lines</td></tr><tr>
<td>7</td><td>decrease graph opacity</td></tr><tr>
<td>9</td><td>increase graph opacity</td></tr><tr>
<td>Shift F3</td><td>toggle population plot</td></tr><tr>
<td>Shift F4</td><td>toggle births plot</td></tr><tr>
<td>Shift F5</td><td>toggle deaths plot</td></tr><tr>
<td>Shift W</td><td>toggle performance warning</td></tr><tr>
<td>?</td><td>toggle cell type for hexagonal and triangular grids</td></tr><tr>
<td>T</td><td>toggle timing information</td></tr><tr>
<td>Shift T</td><td>toggle extended timing information</td></tr><tr>
<td>I</td><td>toggle pattern and engine information</td></tr><tr>
<td>Shift I</td><td>toggle information bar</td></tr><tr>
<td>Ctrl Q</td><td>toggle cell anti-aliasing</td></tr><tr>
<td>Alt B</td><td>toggle cell borders</td></tr><tr>
<td>Alt I</td><td>toggle icons</td></tr><tr>
<td>X</td><td>toggle gridlines</td></tr><tr>
<td>Shift X</td><td>toggle major gridlines</td></tr><tr>
<td>Ctrl G</td><td>toggle auto gridlines</td></tr><tr>
<td>Alt G</td><td>convert pattern to [Rule]Super</td></tr><tr>
<td>Alt H</td><td>convert pattern to [Rule]History</td></tr><tr>
<td>Alt J</td><td>convert pattern to [Rule] (from either [Rule]History or [Rule]Super)</td></tr><tr>
<td>Ctrl L</td><td>toggle kill escaping gliders</td></tr><tr>
<td>S</td><td>toggle stars</td></tr><tr>
<td>O</td><td>save screenshot</td></tr><tr>
<td>Shift O</td><td>save population graph screenshot</td></tr><tr>
<td>;</td><td>save black and white screenshot</td></tr><tr>
<td>Shift U</td><td>toggle UI</td></tr>
<td>Alt U</td><td>toggle autohide UI during playback</td></tr><tr>
<td>Alt W</td><td>cycle cell shader</td></tr><tr>
<td>D</td><td>download cell map</td></tr><tr>
<td>E</td><td>cycle identify results display</td></tr><tr>
<td>E</td><td>toggle cell period table</td></tr>
<td>F8</td><td>toggle cell state display</td></tr>
<td>Shift F8</td><td>toggle y coordinate direction</td></tr>
<td>F9</td><td>toggle popup window maximize</td></tr>
</table>
</div>
</div>

=== Pattern controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Pattern controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Alt N</td><td>new pattern</td></tr><tr>
<td>Alt R</td><td>change rule</td></tr><tr>
<td>Ctrl O</td><td>open original or last saved pattern</td></tr><tr>
<td>Ctrl+Shift O</td><td>open clipboard as pattern</td></tr><tr>
<td>Ctrl S</td><td>save pattern</td></tr><tr>
<td>Alt Z</td><td>randomize pattern and rule</td></tr><tr>
<td>Ctrl+Alt Z</td><td>randomize pattern only</td></tr><tr>
<td>F6</td><td>toggle oscillator and spaceship identification</td></tr><tr>
<td>Shift+F6</td><td>display last identification result</td></tr><tr>
<td>Shift+F9</td><td>toggle safe mode</td></tr>
</table>
</div>
</div>

=== External clipboard controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view External clipboard controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Alt S</td><td>toggle sync cut and copy with external clipboard</td></tr><tr>
<td>Ctrl+Shift C</td><td>copy original pattern</td></tr><tr>
<td>Ctrl C</td><td>copy current selection</td></tr><tr>
<td>Ctrl+Alt C</td><td>copy current selection with comments</td></tr><tr>
<td>F7</td><td>copy selection headerless including blanks</td></tr><tr>
<td>Ctrl F7</td><td>copy selection headerless including blanks (1 line)</td></tr><tr>
<td>Ctrl+Shift X</td><td>cut original pattern</td></tr><tr>
<td>Ctrl X</td><td>cut current selection</td></tr><tr>
<td>Ctrl+Alt X</td><td>cut current selection with comments</td></tr><tr>
<td>Ctrl J</td><td>copy rule definition</td></tr><tr>
<td>K</td><td>copy camera position</td></tr><tr>
<td>Shift K</td><td>copy camera position and view</td></tr>
</table>
</div>
</div>

=== Edit controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Edit controls.
<div class="mw-collapsible-content">
<table border=1 cellspacing=0 width=500><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>F1</td><td>toggle draw/pan mode</td></tr><tr>
<td>F2</td><td>draw mode</td></tr><tr>
<td>Shift F2</td><td>toggle smart drawing</td></tr><tr>
<td>F3</td><td>pick mode</td></tr><tr>
<td>F4</td><td>select mode</td></tr><tr>
<td>F5</td><td>pan mode</td></tr><tr>
<td>Ctrl A</td><td>select all</td></tr><tr>
<td>Shift A</td><td>shrink selection</td></tr><tr>
<td>Alt A</td><td>toggle auto-shrink selection</td></tr><tr>
<td>Ctrl C</td><td>copy</td></tr><tr>
<td>Ctrl D</td><td>toggle states display</td></tr><tr>
<td>Ctrl I</td><td>invert selection</td></tr><tr>
<td>Ctrl K</td><td>remove selection</td></tr><tr>
<td>Ctrl V</td><td>paste</td></tr><tr>
<td>Ctrl+Shift V</td><td>paste to selection</td></tr><tr>
<td>Ctrl X</td><td>cut</td></tr><tr>
<td>Ctrl Y</td><td>redo edit</td></tr><tr>
<td>Ctrl Z</td><td>undo edit</td></tr><tr>
<td>Ctrl+Shift Z</td><td>redo edit</td></tr><tr>
<td>Ctrl Space</td><td>advance selection</td></tr><tr>
<td>Shift Space</td><td>advance outside</td></tr><tr>
<td>Shift L</td><td>cycle paste location</td></tr><tr>
<td>Shift M</td><td>cycle paste mode</td></tr><tr>
<td>Shift 5</td><td>random fill</td></tr><tr>
<td>Alt K</td><td>pick cell state to replace with drawing state</td></tr><tr>
<td>Ctrl+Alt K</td><td>clear current drawing state cells</td></tr><tr>
<td>Del</td><td>clear cells in selection</td></tr><tr>
<td>Shift Del</td><td>clear outside selection</td></tr><tr>
<td>></td><td>rotate selection clockwise</td></tr><tr>
<td><</td><td>rotate selection counter-clockwise</td></tr><tr>
<td>Alt X</td><td>flip selection horizontally</td></tr><tr>
<td>Alt Y</td><td>flip selection vertically</td></tr><tr>
<td>Alt Left</td><td>nudge selection 1 cell left</td></tr><tr>
<td>Alt Right</td><td>nudge selection 1 cell right</td></tr><tr>
<td>Alt Up</td><td>nudge selection 1 cell up</td></tr><tr>
<td>Alt Down</td><td>nudge selection 1 cell down</td></tr><tr>
<td>Esc</td><td>hide paste</td></tr><tr>
<td>Enter</td><td>perform paste</td></tr><tr>
<td>Ctrl 0</td><td>select state 0 for drawing</td></tr><tr>
<td>Ctrl 1</td><td>select state 1 for drawing</td></tr><tr>
<td>Shift B</td><td>toggles clipboard library</td></tr><tr>
<td>Alt 0</td><td>make clipboard 0 active</td></tr><tr>
<td>Alt 1</td><td>make clipboard 1 active</td></tr><tr>
<td>Alt 2</td><td>make clipboard 2 active</td></tr><tr>
<td>Alt 3</td><td>make clipboard 3 active</td></tr><tr>
<td>Alt 4</td><td>make clipboard 4 active</td></tr><tr>
<td>Alt 5</td><td>make clipboard 5 active</td></tr><tr>
<td>Alt 6</td><td>make clipboard 6 active</td></tr><tr>
<td>Alt 7</td><td>make clipboard 7 active</td></tr><tr>
<td>Alt 8</td><td>make clipboard 8 active</td></tr><tr>
<td>Alt 9</td><td>make clipboard 9 active</td></tr>
</table>
</div>
</div>

=== Help controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view Help controls.
<div class="mw-collapsible-content">
<td><table border=1 cellspacing=0 width=500><tr></td></tr><tr>
<td bgcolor=#c0c0c0>Key</td><td bgcolor=#c0c0c0>Description</td></tr><tr>
<td>Up</td><td>scroll up one line</td></tr><tr>
<td>Down</td><td>scroll down one line</td></tr><tr>
<td>Page Up</td><td>scroll up one page</td></tr><tr>
<td>Page Down</td><td>scroll down one page</td></tr><tr>
<td>Shift Page Up</td><td>scroll up one section</td></tr><tr>
<td>Shift Page Down</td><td>scroll down one section</td></tr><tr>
<td>Home</td><td>go to first help page</td></tr><tr>
<td>End</td><td>go to last help page</td></tr>
</table>
</div>
</div>

=== Controls without keyboard shortcuts ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view functions that do not currently have keyboard shortcuts.
<div class="mw-collapsible-content">
{| class="wikitable" style="margin-left: auto; margin-right: auto; text-align: center;"
! Command
! Button location
|-
! Random Fill Density
| Select; a selection is present
|-
! individual Themes
| Settings --> Theme
|-
! Draw/2-state Fill with states 10-255
| Draw
|}
</div>
</div>

=== Keyboard shortcuts without controls ===
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view functions with keyboard shortcuts but no UI controls.
<div class="mw-collapsible-content">
{| class="wikitable" style="margin-left: auto; margin-right: auto; text-align: center;"
! Command
! Shortcut
|-
! [R]Super
| Alt + G
|-
! [R]History
| Alt + H
|-
! [R]Standard
| Alt + J
|}
</div>
</div>
[[Category:Software]]
[[Category:Everything else]]

LifeViewer/User Interface

2026-02-28T09:14:41Z

Remu:

Comparison of oscillator analysis tools

2026-02-28T09:13:37Z

Remu: Category:Software

This page compares tools used for analyzing [[still life]]s, [[oscillator]]s and [[spaceship]]s alongside the information they output.

We assume that the input pattern runs in B3/S23, as these tools are used for computing information for wiki infoboxes.

== Oscillators ==
{| class="wikitable" style="margin-left: auto; margin-right: auto; text-align: center"
! colspan="3" | Datum
! [[Oscar]]
! [[Oscillizer]]
! [[Oscillizer-Web]]
! [[Oscilloscope]]
! [[Nakano]]
! [[LifeViewer]]
! [[CAViewer]]
! [[Caterer (program)|Caterer]]
! [[Catagolue]]
|-
! colspan="3" | [[Period]]
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''≤1200'''
| style="background-color:#c6efce; color:#006100" | '''≤1000'''
| style="background-color:#c6efce; color:#006100" | '''≤50000'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''<ref group="n">no upper limit as of build 1285 (possibly earlier)</ref>
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! rowspan="2" | [[Mod]]
! colspan="2" | Value
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Transformation
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="3" | [[Kinetic symmetry]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="3" | [[Population]]
! colspan="2" | Minimum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="2" | Maximum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="2" | Average
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="3" | Active cells
! colspan="2" | [[Rotor]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | [[Stator]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Total
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | [[Volatility]]
! colspan="2" | Standard
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Strict
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="3" | [[Heat]]
! colspan="2" | Minimum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Maximum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Average
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | [[Temperature]]
! colspan="2" | Active
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Rotor
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="6" | [[Bounding box]]
! rowspan="2" | All phases
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | Min. phase
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | Max. phase
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="4" | Supported<br>by rules
! rowspan="2" | Minimum
! OT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! INT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! rowspan="2" | Maximum
! OT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! INT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="3" | [[apgcode]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|}

=== Mapping ===
{| class="wikitable" style="margin-left: auto; margin-right: auto; text-align: center;"
! Map type
! [[Oscar]]
! [[Oscillizer]]
! [[Oscillizer-Web]]
! [[Oscilloscope]]
! [[Nakano]]
! [[LifeViewer]]
! [[CAViewer]]
! [[Caterer (program)|Caterer]]
! [[Catagolue]]
|-
! [[Period map]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! [[Frequency map]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! [[Heat map]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! [[Signature map]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! [[Phase coloring]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|}

== Spaceships ==
{| class="wikitable" style="margin-left: auto; margin-right: auto; text-align: center"
! colspan="3" | Datum
! [[Oscar]]
! [[Oscilloscope]]
! [[Nakano]]
! [[LifeViewer]]
! [[CAViewer]]
! [[Caterer (program)|Caterer]]
! [[Catagolue]]
|-
! colspan="3" | [[Period]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! rowspan="2" | [[Mod]]
! colspan="2" | Value
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Transformation
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="3" | [[Kinetic symmetry]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Glide'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="3" | [[Population]]
! colspan="2" | Minimum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="2" | Maximum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="2" | Average
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="3" | [[Direction]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffffdd; color:#616100" | '''dx, dy'''
| style="background-color:#ffffdd; color:#616100" | '''dx, dy'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="3" | [[Slope]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | [[Speed]]
! colspan="2" | Unsimplified
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Simplified
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="3" | [[Heat]]
! colspan="2" | Minimum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Maximum
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! colspan="2" | Average
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="6" | [[Bounding box]]
! rowspan="2" | All phases
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | Min. phase
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="2" | Max. phase
! XY
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! Area
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffffdd; color:#616100" | '''Help only'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! rowspan="4" | Supported by rules
! rowspan="2" | Minimum
! OT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! INT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! rowspan="2" | Maximum
! OT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
|-
! INT
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|-
! colspan="3" | [[apgcode]]
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#ffc7ce; color:#9c0006" | '''No'''
| style="background-color:#c6efce; color:#006100" | '''Yes'''
|}

== Example maps ==
The [[p96 Hans Leo hassler]] is used for these demos due to its low period and diverse set of subperiods. Legends are not included.

LifeViewer does not display the initial pattern inside the period map as to do so would be redundant given that the input pattern is already accessible. In Oscilloscope, the pattern preview is animated.

=== Period maps ===
<gallery>
p96 Hans Leo hassler period map (Oscillizer).png|Oscillizer
p96 Hans Leo hassler period map (Oscillizer-Web).png|Oscillizer-Web
p96 Hans Leo hassler period map (Oscillizer-Web with interiors).png|Oscillizer-Web, interiors enabled
p96 Hans Leo hassler period map (Oscillizer-Web with borders).png|Oscillizer-Web, borders enabled
p96 Hans Leo hassler period map (Nakano).png|Nakano
</gallery><br>
<gallery>
p96 Hans Leo hassler period map (Oscilloscope v1.1.0).png|Oscilloscope (v1.1.0)
p96 Hans Leo hassler period map (Oscilloscope v1.2.0).png|Oscilloscope (v1.2.0)
p96 Hans Leo hassler period map (Oscilloscope v1.2.0 grayscale).png|Oscilloscope (v1.2.0, grayscale)
</gallery><br>
<gallery>
p96 Hans Leo hassler period map (LifeViewer build 804).png|LifeViewer (builds 804-808)
p96 Hans Leo hassler period map (LifeViewer build 809).png|LifeViewer (builds 809-814)
p96 Hans Leo hassler period map (LifeViewer build 815).png|LifeViewer (build 815)
p96 Hans Leo hassler period map (LifeViewer build 816).png|LifeViewer (builds 816-942)
p96 Hans Leo hassler period map (LifeViewer build 943).png|LifeViewer (builds 943-956)
p96 Hans Leo hassler period map (LifeViewer build 957).png|LifeViewer (builds 957-969)
p96 Hans Leo hassler period map (LifeViewer build 970).png|LifeViewer (build 970)
p96 Hans Leo hassler period map (LifeViewer build 971).png|LifeViewer (builds 971-1322)
p96 Hans Leo hassler period map (LifeViewer build 1323).png|LifeViewer (builds 1323-1370)
p96 Hans Leo hassler period map (LifeViewer build 1371).png|LifeViewer (builds 1371-present)
</gallery>

=== Frequency maps ===
<gallery>
p96 Hans Leo hassler frequency map (Oscilloscope v1.1.0).png|Oscilloscope (v1.1.0)
p96 Hans Leo hassler frequency map (Oscilloscope v1.2.0).png|Oscilloscope (v1.2.0)
p96 Hans Leo hassler frequency map (Oscilloscope v1.2.0 grayscale).png|Oscilloscope (v1.2.0, grayscale)
</gallery><br>
<gallery>
p96 Hans Leo hassler frequency map (LifeViewer build 1250).png|LifeViewer (build 1250)
p96 Hans Leo hassler frequency map (LifeViewer build 1251).png|LifeViewer (builds 1251-1253)
p96 Hans Leo hassler frequency map (LifeViewer build 1254).png|LifeViewer (builds 1254-present)
</gallery>

=== Heat maps ===
<gallery>
p96 Hans Leo hassler heat map (Oscilloscope v1.1.0).png|Oscilloscope (v1.1.0)
</gallery>

=== Signature maps ===
<gallery>
p96 Hans Leo hassler signature map (Oscilloscope v1.5.0).png|Oscilloscope (v1.5.0)
</gallery>

== Notes ==
<references group="n"/>
[[Category:Software]]

Cubic grid symmetries

2026-02-28T09:12:05Z

Remu: Category:Everything else

An enumeration of the 92 [[cubic grid]] symmetries.

== Notation ==
We will use [https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups Conway notation] for the underlying symmetry group on Euclidean three space. This means the 'subscript' corresponds to the order of the group unless the group name starts with '''±''', in which case the group order is twice the subscript.

We will use the letters '''c''', '''f''', '''e''', and '''k''' to refer to a cell, face, edge, or vertex respectively.

There are six relevant axes, they will be referred to by the two types of facets through whose centers they pass.

For dihedral-type symmetries the first letter will refer to the fixed point, and if necessary to disambiguate the second will give the direction of the axis.

For some pyramidal symmetries '''o''' and '''d''' will refer to an orthogonal or diagonal respectively alignment of the base.

For the three relevant planes: '''c''' will refer to the orthogonal plane through the centers of cells, '''f''' will refer to the orthogonal plane through the faces, and '''/''' will refer to the diagonal plane.

== Symmetries fixing a cell ==

=== Chiral symmetries ===

{| class="wikitable"
! Name
! Description
! Example RLE
|-
| '''O24_c'''
| chiral cubic symmetry centered on a cell
|
<nowiki>
3D version=1 size=13 pos=3,3,3
#C O24_c
x=7 y=7 z=7 rule=3D0..6/F
$4bo$bo2$5bo$2bo/2bo2$3bo2bo$2bobo$o2bo2$4bo/5bo$o2bo2$bo3bo2$3bo2bo$bo/$2bobo$bo3bo2$bo3bo$2bobo/bo$3bo2bo2$bo
3bo2$o2bo$5bo/4bo2$o2bo$2bobo$3bo2bo2$2bo/$2bo$5bo2$bo$4bo!</nowiki>
|-
| '''T12_c'''
| chiral tetrahedral symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,3
#C T12_c
x=5 y=5 z=5 rule=3D0..6/F
$2bo2$2bo/$3bo$o3bo$bo/bobo2$2bo2$bobo/$bo$o3bo$3bo/$2bo2$2bo!</nowiki>
|-
| '''D8_c'''
| chiral square prismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C D8_c
x=5 y=5 z=3 rule=3D0..6/F
bo$4bo2$o$3bo/$2bo$bobo$2bo/3bo$o2$4bo$bo!</nowiki>
|-
| '''C4_cf'''
| chiral square pyramidal symmetry,
a.k.a., 90 degree rotation (line goes through cells and faces)
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C C4_cf
x=5 y=5 z=2 rule=3D0..6/F
$2bo$bobo$2bo/3bo$o2$4bo$bo!</nowiki>
|-
| '''D4_cf'''
| chiral brick symmetry centered on a cell, orthogonal orientation
|
<nowiki>
3D version=1 size=11 pos=4,3,4
#C D4_cf
x=3 y=5 z=3 rule=3D0..6/F
$2bo2$o/bo2$obo2$bo/$o2$2bo!</nowiki>
|-
| '''D4_ce'''
| chiral brick symmetry centered on a cell, diagonal orientation
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C D4_ce
x=5 y=5 z=3 rule=3D0..6/F
bo4$3bo/$bo$2bo$3bo/$o2$4bo!</nowiki>
|-
| '''D6_ck''' or simply '''D6_c'''
| chiral triangular prismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C D6_c
x=3 y=3 z=3 rule=3D0..6/F
2$bo/2bo$bo/$o!</nowiki>
|-
| '''C3_ck''' or simply '''C3'''
| chiral triangular pyramidal symmetry,
a.k.a., 120 degree rotation (line goes through vertices and cells)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C C3
x=3 y=3 z=3 rule=3D0..6/F
2$bo/2bo$2bo$bo/$2o!</nowiki>
|-
| '''C2_cf'''
| chiral digonal pyramidal symmetry,
a.k.a., 180 degree rotation (line goes through cells and faces)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C C2_cf
x=3 y=2 z=3 rule=3D0..6/F
2o/$bo/b2o!</nowiki>
|-
| '''C2_ce'''
| chiral digonal pyramidal symmetry,
a.k.a., 180 degree rotation (line goes through cells and edges)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C C2_ce
x=2 y=2 z=3 rule=3D0..6/F
o/$o/$bo!</nowiki>
|-
| '''C1'''
| no symmetry
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C C1
x=2 y=3 z=3 rule=3D0..6/F
o/$o/$bo$bo!</nowiki>
|}

=== Containing the inversion ===

{| class="wikitable"
! Name
! Description
! Example RLE
|-
| '''±O24_c'''
| cubic symmetry centered on a cell
|
<nowiki>
3D version=1 size=7 pos=3,3,3
#C ±O24_c
x=1 y=1 z=1 rule=3D0..6/F
o!</nowiki>
|-
| '''±T12_c'''
| pyritic symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,3
#C ±T12_c
x=5 y=5 z=5 rule=3D0..6/F
2$bobo/2bo2$2bo2$2bo/$obobo$bobo$obobo/2bo2$2bo2$2bo/2$bobo!</nowiki>
|-
| '''±D8_c'''
| square prismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=9 pos=4,4,3
#C ±D8_c
x=1 y=1 z=3 rule=3D0..6/F
o/o/o!</nowiki>
|-
| '''±C4_cf'''
| square proprismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,5
#C ±C4_cf
x=5 y=5 z=1 rule=3D0..6/F
3bo$obo$bobo$2bobo$bo!</nowiki>
|-
| '''±D4_cf'''
| brick symmetry centered on a cell, orthogonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,4
#C ±D4_cf
x=3 y=3 z=1 rule=3D0..6/F
obo$3o$obo!</nowiki>
|-
| '''±D4_ce'''
| brick symmetry centered on a cell, diagonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,4
#C ±D4_ce
x=3 y=3 z=1 rule=3D0..6/F
o$bo$2bo!</nowiki>
|-
| '''±D6_ck''' or simply '''±D6_c'''
| triangular antiprismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±D6_c
x=3 y=3 z=3 rule=3D0..6/F
o/$bo/2$2bo!</nowiki>
|-
| '''±C3_ck''' or simply '''±C3_c'''
| triangular proantiprismatic symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,3
#C ±C3_c
x=5 y=5 z=5 rule=3D0..6/F
$2bo/$bo$o/bo2$2bo2$3bo/2$4bo$3bo/3$2bo!</nowiki>
|-
| '''±C2_cf'''
| espic symmetry centered on a cell, axis goes through face
|
<nowiki>
3D version=1 size=9 pos=3,4,3
#C ±C2_cf
x=3 y=1 z=3 rule=3D0..6/F
o/3o/2bo!</nowiki>
|-
| '''±C2_ce'''
| espic symmetry centered on a cell, axis goes through edge
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C2_ce
x=3 y=3 z=3 rule=3D0..6/F
o/o$bo$2bo/2$2bo!</nowiki>
|-
| '''±C1_c'''
| central inversion symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=4,4,3
#C ±C1_c
x=3 y=3 z=5 rule=3D0..6/F
o/$o/$bo/$2bo/2$2bo!</nowiki>
|}

=== Other achiral symmetries ===

{| class="wikitable"
! Name
! Description
! Example RLE
|-
| '''TO24_c'''
| tetrahedral symmetry centered on a cell
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C TO24_c
x=3 y=3 z=3 rule=3D0..6/F
2bo2$o/$bo/o2$2bo!</nowiki>
|-
| '''CD6_ck''' or simply '''CD6'''
| triangular pyramidal symmetry (line goes through vertices and cells)
|
<nowiki>
3D version=1 size=8 pos=3,3,3
#C CD6
x=2 y=2 z=2 rule=3D0..6/F
2o$o/o!</nowiki>
|-
| '''CD8_cf'''
| square pyramidal symmetry (line goes through cells and faces)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C CD8_cf
x=3 y=3 z=2 rule=3D0..6/F
$bo/bo$obo$bo!</nowiki>
|-
| '''DD8_cf'''
| digonal antiprismatic symmetry centered on a cell, digon is diagonal
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C DD8_cf
x=3 y=3 z=3 rule=3D0..6/F
2bo$bo$o/$bo/o$bo$2bo!</nowiki>
|-
| '''DD8_ce'''
| digonal antiprismatic symmetry centered on a cell, digon is orthogonal
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C DD8_ce
x=3 y=3 z=3 rule=3D0..6/F
bo2$bo/$bo/$obo!</nowiki>
|-
| '''CC4_cf'''
| prodappic symmetry centered on a cell
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C CC4_cf
x=5 y=5 z=3 rule=3D0..6/F
3bo4$bo/$2bo$bobo$2bo/$o2$4bo!</nowiki>
|-
| '''CD4_cfo'''
| digonal pyramidal symmetry (line goes through cells and faces), orthogonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,4
#C CD4_cfo
x=3 y=2 z=1 rule=3D0..6/F
bo$obo!</nowiki>
|-
| '''CD4_cfd'''
| digonal pyramidal symmetry (line goes through cells and faces), diagonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C CD4_cfd
x=3 y=3 z=2 rule=3D0..6/F
$bo/o2$2bo!</nowiki>
|-
| '''CD4_ce'''
| digonal pyramidal symmetry (line goes through cells and edges)
|
<nowiki>
3D version=1 size=8 pos=3,3,3
#C CD4_ce
x=2 y=2 z=1 rule=3D0..6/F
2o$o!</nowiki>
|-
| '''CC2_c'''
| reflection (plane goes through cells, edges, and faces)
|
<nowiki>
3D version=1 size=9 pos=4,3,3
#C CC2_c
x=1 y=3 z=2 rule=3D0..6/F
o$o/2$o!</nowiki>
|-
| '''CC2_/'''
| reflection (plane goes through cells, faces, edges, and vertices)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C CC2_/
x=3 y=3 z=2 rule=3D0..6/F
o/$bo$2bo!</nowiki>
|}

== Symmetries fixing a face but not a cell ==

=== Chiral symmetries ===

{| class="wikitable"
! Name
! Description
! Example RLE
|-
| '''D8_f'''
| chiral square prismatic symmetry centered on a face
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C D8_f
x=5 y=5 z=2 rule=3D0..6/F
bo$2bobo$bobo$obo$3bo/3bo$obo$bobo$2bobo$bo!</nowiki>
|-
| '''D4_fe'''
| chiral brick symmetry centered on a face, orthogonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C D4_fe
x=3 y=3 z=2 rule=3D0..6/F
2bo$3o$o/o$3o$2bo!</nowiki>
|-
| '''D4_fk'''
| chiral brick symmetry centered on a face, diagonal orientation
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C D4_fk
x=5 y=5 z=2 rule=3D0..6/F
$o$bobo$4bo/bo$2bo2$2bo$3bo!</nowiki>
|-
| '''C2_fe'''
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and edges)
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C C2_fe
x=3 y=2 z=2 rule=3D0..6/F
bo$2bo/bo$o!</nowiki>
|-
| '''C2_fk'''
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and vertices)
|
<nowiki>
3D version=1 size=8 pos=3,3,3
#C C2_fk
x=2 y=2 z=2 rule=3D0..6/F
2o/o$o!</nowiki>
|}

=== Containing the inversion ===

{| class="wikitable"
! Name
! Description
! Example RLE
|-
| '''±D8_f'''
| square prismatic symmetry centered on a face
|
<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±D8_f
x=1 y=1 z=2 rule=3D0..6/F
o/o!</nowiki>
|-
| '''±D4_fe'''
| brick symmetry centered on a face, orthogonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,4,3
#C ±D4_fe
x=3 y=1 z=2 rule=3D0..6/F
3o/3o!</nowiki>
|-
| '''±D4_fk'''
| brick symmetry centered on a face, diagonal orientation
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±D4_fk
x=3 y=3 z=2 rule=3D0..6/F
o$bo$2bo/o$bo$2bo!</nowiki>
|-
| '''±C4_fc'''
| square proprismatic symmetry centered on a face
|
<nowiki>
3D version=1 size=11 pos=3,3,4
#C ±C4_fc
x=5 y=5 z=2 rule=3D0..6/F
3bo$obo$bobo$2bobo$bo/3bo$obo$bobo$2bobo$bo!</nowiki>
|-
| '''±C2_fc'''
| espic symmetry centered on a face, axis goes through cell
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C2_fc
x=3 y=3 z=2 rule=3D0..6/F
o$3o$2bo/o$3o$2bo!</nowiki>
|-
| '''±C2_fe'''
| espic symmetry centered on a face, axis goes through edge
|
<nowiki>
3D version=1 size=9 pos=4,3,3
#C ±C2_fe
x=1 y=3 z=2 rule=3D0..6/F
o$o/$o$o!</nowiki>
|-
| '''±C2_fk'''
| espic symmetry centered on a face, axis goes through vertex
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C2_fk
x=3 y=3 z=2 rule=3D0..6/F
$bo$2bo/o$bo!</nowiki>
|-
| '''±C1_f'''
| central inversion symmetry centered on a face
|
<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C1_f
x=3 y=3 z=2 rule=3D0..6/F
$b2o$2bo/o$2o!</nowiki>
|}

=== Other achiral symmetries ===

'''DD8_fe''': digonal antiprismatic symmetry centered on a face, digon is diagonal

<nowiki>
3D version=1 size=9 pos=3,3,3
#C DD8_fe
x=3 y=3 z=2 rule=3D0..6/F
o$bo$2bo/2bo$bo$o!</nowiki>

'''DD8_fk''': digonal antiprismatic symmetry centered on a face, digon is orthogonal

<nowiki>
3D version=1 size=9 pos=3,3,3
#C DD8_fk
x=3 y=3 z=2 rule=3D0..6/F
$obo/bo2$bo!</nowiki>

'''CC4_fc''': prodappic symmetry centered on a face

<nowiki>
3D version=1 size=10 pos=3,3,3
#C CC4_fc
x=3 y=3 z=4 rule=3D0..6/F
2bo2$o/$obo/bo2$bo/o2$2bo!</nowiki>

'''CD4_fe''': digonal pyramidal symmetry (line goes through faces and edges)

<nowiki>
3D version=1 size=10 pos=4,4,3
#C CD4_fe
x=1 y=2 z=4 rule=3D0..6/F
$o/o/o/$o!</nowiki>

'''CD4_fk''': digonal pyramidal symmetry (line goes through faces and vertices)

<nowiki>
3D version=1 size=10 pos=4,4,3
#C CD4_fk
x=2 y=2 z=4 rule=3D0..6/F
$bo/o/o/$bo!</nowiki>

'''CC2_f''': reflection (plane goes through faces, edges, and vertices)

<nowiki>
3D version=1 size=9 pos=3,3,3
#C CC2_f
x=3 y=2 z=2 rule=3D0..6/F
2o$2bo/2o$2bo!</nowiki>

== Symmetries fixing an edge but not a cell or face ==

=== Chiral symmetries ===

'''D8_e''': chiral square prismatic symmetry centered on an edge

<nowiki>
3D version=1 size=10 pos=3,3,3
#C D8_e
x=3 y=4 z=4 rule=3D0..6/F
$o$2bo/2bo$bo$bo$o/o$bo$bo$2bo/$2bo$o!</nowiki>

'''D4_ef''': chiral brick symmetry centered on an edge, orthogonal orientation

<nowiki>
3D version=1 size=10 pos=3,3,4
#C D4_ef
x=3 y=4 z=2 rule=3D0..6/F
2bo$bo$bo$o/o$bo$bo$2bo!</nowiki>

'''D4_ec''': chiral brick symmetry centered on an edge, diagonal orientation

<nowiki>
3D version=1 size=10 pos=3,3,3
#C D4_ec
x=3 y=4 z=4 rule=3D0..6/F
$o/2bo$bo/2$bo$2bo/2$o!</nowiki>

'''C4_ek''': chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through edges and vertices)

<nowiki>
3D version=1 size=10 pos=4,3,3
#C C4_ek
x=2 y=4 z=4 rule=3D0..6/F
2$bo/bo$o$o/$o$o$bo/$bo!</nowiki>

'''C2_ek''': chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through edges and vertices)

<nowiki>
3D version=1 size=10 pos=4,4,3
#C C2_ek
x=2 y=2 z=4 rule=3D0..6/F
o/bo/$bo/$o!</nowiki>

=== Containing the inversion ===

'''±D8_e''': square prismatic symmetry centered on an edge

<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±D8_e
x=1 y=2 z=2 rule=3D0..6/F
o$o/o$o!</nowiki>

'''±D4_ef''': brick symmetry centered on an edge, orthogonal orientation

<nowiki>
3D version=1 size=10 pos=4,3,4
#C ±D4_ef
x=1 y=4 z=2 rule=3D0..6/F
o$o$o$o/o$o$o$o!</nowiki>

'''±D4_ec''': brick symmetry centered on an edge, diagonal orientation

<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±D4_ec
x=1 y=2 z=2 rule=3D0..6/F
o/$o!</nowiki>

'''±C4_ek''': square proprismatic symmetry centered on an edge

<nowiki>
3D version=1 size=10 pos=4,3,3
#C ±C4_ek
x=1 y=4 z=4 rule=3D0..6/F
2$o/o$o$o/$o$o$o/$o!</nowiki>

'''±C2_ek''': espic symmetry centered on an edge, axis goes through vertex

<nowiki>
3D version=1 size=10 pos=4,3,4
#C ±C2_ek
x=1 y=4 z=2 rule=3D0..6/F
o$o/2$o$o!</nowiki>

'''±C2_ef''': espic symmetry centered on an edge, axis goes through face

<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C2_ef
x=3 y=2 z=2 rule=3D0..6/F
b2o$2o/b2o$2o!</nowiki>

'''±C2_ec''': espic symmetry centered on an edge, axis goes through cell

<nowiki>
3D version=1 size=9 pos=3,3,3
#C ±C2_ec
x=3 y=2 z=2 rule=3D0..6/F
b2o/$2o!</nowiki>

'''±C1_e''': central inversion symmetry centered on an edge

<nowiki>
3D version=1 size=10 pos=3,3,4
#C ±C1_e
x=3 y=4 z=2 rule=3D0..6/F
2bo$bo/2$bo$o!</nowiki>

=== Other achiral symmetries ===

'''DD8_ef''': digonal antiprismatic symmetry centered on an edge, digon is diagonal

<nowiki>
3D version=1 size=9 pos=3,3,3
#C DD8_ef
x=3 y=2 z=2 rule=3D0..6/F
b2o$2o/2o$b2o!</nowiki>

'''DD8_ec''': digonal antiprismatic symmetry centered on an edge, digon is orthogonal

<nowiki>
3D version=1 size=10 pos=3,3,3
#C DD8_ec
x=3 y=4 z=4 rule=3D0..6/F
$o$o/2bo$bo$bo$2bo/2bo$bo$bo$2bo/$o$o!</nowiki>

'''CD8_ek''': square pyramidal symmetry (line goes through edges and vertices)

<nowiki>
3D version=1 size=10 pos=4,3,3
#C CD8_ek
x=2 y=4 z=4 rule=3D0..6/F
$bo$bo/bo$o$o$bo/bo$o$o$bo/$bo$bo!</nowiki>

'''CC4_ek''': prodappic symmetry centered on an edge

<nowiki>
3D version=1 size=10 pos=3,3,3
#C CC4_ek
x=3 y=4 z=4 rule=3D0..6/F
2$o/2bo$bo$bo/$bo$bo$2bo/$o!</nowiki>

'''CD4_eko''': digonal pyramidal symmetry (line goes through edges and vertices), orthogonal orientation

<nowiki>
3D version=1 size=10 pos=4,3,4
#C CD4_eko
x=2 y=4 z=2 rule=3D0..6/F
bo$o$o$bo/bo$o$o$bo!</nowiki>

'''CD4_ekd''': digonal pyramidal symmetry (line goes through edges and vertices), diagonal orientation

<nowiki>
3D version=1 size=10 pos=4,3,3
#C CD4_ekd
x=2 y=4 z=4 rule=3D0..6/F
bo/$o/2$o/3$bo!</nowiki>

== Symmetries fixing a vertex but not an edge, face, or cell==

=== Chiral symmetries ===

'''O24_k''': chiral cubic symmetry centered on a vertex

<nowiki>
3D version=1 size=12 pos=3,3,3
#C O24_k
x=6 y=6 z=6 rule=3D0..6/F
$2bo$4bo$bo$3bo/3bo$bo2bo$o$5bo$bo2bo$2bo/bo$5bo3$o$4bo/4bo$o3$5bo$bo/2bo$bo2bo$5bo$o$bo2bo$3bo/$3bo$bo$4bo$2bo!</nowiki>

'''T12_k''': chiral tetrahedral symmetry centered on a vertex

<nowiki>
3D version=1 size=12 pos=3,3,3
#C T12_k
x=6 y=6 z=6 rule=3D0..6/F
$3bo3$2bo/2$3bobo$obo/4bo$3bo$4bo$bo$2bo$bo/bo$2bo$bo$4bo$3bo$4bo/2$obo$3bobo/$2bo3$3bo!</nowiki>

'''D8_k''': chiral square prismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,3,3
#C D8_k
x=4 y=4 z=4 rule=3D0..6/F
2bo$o$3bo$bo/$b2o$b2o/$b2o$b2o/bo$3bo$o$2bo!</nowiki>

'''D4_ke''': chiral brick symmetry centered on a vertex, orthogonal orientation

<nowiki>
3D version=1 size=12 pos=4,3,5
#C D4_ke
x=4 y=6 z=2 rule=3D0..6/F
bo$o$o$3bo$3bo$2bo/2bo$3bo$3bo$o$o$bo!</nowiki>

'''D4_kf''': chiral brick symmetry centered on a vertex, diagonal orientation

<nowiki>
3D version=1 size=10 pos=3,3,4
#C D4_kf
x=4 y=4 z=2 rule=3D0..6/F
$2o$2b2o/bo$bo$2bo$2bo!</nowiki>

'''D6_kc''' or simply '''D6_k''': chiral triangular prismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,3,3
#C D6_k
x=4 y=4 z=4 rule=3D0..6/F
2$bo/2bo$bo2$2bo/$o2bo$2bo/2$bo!</nowiki>

=== Containing the inversion ===

'''±O24_k''': cubic symmetry centered on a vertex

<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±O24_k
x=2 y=2 z=2 rule=3D0..6/F
2o$2o/2o$2o!</nowiki>

'''±T12_k''': pyritic symmetry centered on a vertex

<nowiki>
3D version=1 size=12 pos=3,3,3
#C ±T12_k
x=6 y=6 z=6 rule=3D0..6/F
2$bo2bo$bo2bo/2b2o$bo2bo3$bo2bo$2b2o/$o4bo3$o4bo/$o4bo3$o4bo/2b2o$bo2bo3$bo2bo$2b2o/2$bo2bo$bo2bo!</nowiki>

'''±D8_k''': square prismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=4,4,3
#C ±D8_k
x=2 y=2 z=4 rule=3D0..6/F
2o$2o/2o$2o/2o$2o/2o$2o!</nowiki>

'''±C4_ke''': square proprismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,3,4
#C ±C4_ke
x=4 y=4 z=2 rule=3D0..6/F
2bo$3o$b3o$bo/2bo$3o$b3o$bo!</nowiki>

'''±D4_ke''': brick symmetry centered on a vertex, orthogonal orientation

<nowiki>
3D version=1 size=10 pos=3,3,4
#C ±D4_ke
x=4 y=4 z=2 rule=3D0..6/F
o2bo$4o$4o$o2bo/o2bo$4o$4o$o2bo!</nowiki>

'''±D4_kf''': brick symmetry centered on a vertex, diagonal orientation

<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±D4_kf
x=2 y=2 z=2 rule=3D0..6/F
o$bo/o$bo!</nowiki>

'''±D6_kc''' or simply '''±D6_k''': triangular antiprismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=8 pos=3,3,3
#C ±D6_k
x=2 y=2 z=2 rule=3D0..6/F
o/$bo!</nowiki>

'''±C3_kc''' or simply '''±C3_k''': triangular proantiprismatic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,3,3
#C ±C3_k
x=4 y=4 z=4 rule=3D0..6/F
2$bo/2bo$bo$3bo/$o$2bo$bo/$2bo!</nowiki>

'''±C2_ke''': espic symmetry centered on a vertex, axis goes through edge

<nowiki>
3D version=1 size=10 pos=3,4,4
#C ±C2_ke
x=4 y=2 z=2 rule=3D0..6/F
2o$2o/2b2o$2b2o!</nowiki>

'''±C2_kf''': espic symmetry centered on a vertex, axis goes through face

<nowiki>
3D version=1 size=10 pos=4,4,3
#C ±C2_kf
x=2 y=2 z=4 rule=3D0..6/F
o/o/$bo/$bo!</nowiki>

'''±C1_k''': central inversion symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,4,4
#C ±C1_k
x=4 y=2 z=2 rule=3D0..6/F
o$bo/2bo$3bo!</nowiki>

=== Other achiral symmetries ===

'''TO24_k''': tetrahedral symmetry centered on a vertex

<nowiki>
3D version=1 size=8 pos=3,3,3
#C TO24_k
x=2 y=2 z=2 rule=3D0..6/F
o$bo/bo$o!</nowiki>

'''DD8_ke''': digonal antiprismatic symmetry centered on a vertex, digon is diagonal

<nowiki>
3D version=1 size=10 pos=4,4,3
#C DD8_ke
x=2 y=2 z=4 rule=3D0..6/F
o$bo/o$bo/bo$o/bo$o!</nowiki>

'''DD8_kf''': digonal antiprismatic symmetry centered on a vertex, digon is orthogonal

<nowiki>
3D version=1 size=10 pos=3,3,4
#C DD8_kf
x=4 y=4 z=2 rule=3D0..6/F
$o2bo$o2bo/b2o3$b2o!</nowiki>

'''CC4_ke''': prodappic symmetry centered on a vertex

<nowiki>
3D version=1 size=10 pos=3,3,3
#C CC4_ke
x=4 y=4 z=4 rule=3D0..6/F
$o$3bo/$bo$2bo/$2bo$bo/2bo3$bo!</nowiki>

== Generators ==
Below is a list of generators for these groups.

Each row contains the group name, followed by the list of generators.

Each generator consists of a {{times|3|3}} matrix, M, followed by a length 3 vector, v. The transformation is M*'''x'''+v where '''x''' is the coordinates of the point.

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view the generators data.
<div class="mw-collapsible-content">
<nowiki>
[
['C1', []],
['C2_cf', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]],
['C2_ce', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]],
['CC2_/', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±C1_c', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]],
['CC2_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]],
['C3', [[[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]],
['D4_cf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]],
['C4_cf', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['CC4_cf', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]],
['CD4_cfd', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['CD4_ce', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]],
['±C2_ce', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±C2_cf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]],
['CD4_cfo', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]],
['D4_ce', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['CD6', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]],
['D6_c', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]],
['±C3_c', [[[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [0, 0, 0]]]],
['±D4_cf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]],
['±C4_cf', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]],
['±D4_ce', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['DD8_ce', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]],
['DD8_cf', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]],
['CD8_cf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['D8_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±D6_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [0, 0, 0]]]],
['T12_c', [[[[0, 0, -1], [-1, 0, 0], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]],
['±D8_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]],
['±T12_c', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [-1, 0, 0]], [0, 0, 0]]]],
['TO24_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 0, 0]]]],
['O24_c', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 0]]]],
['±O24_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 0]]]],
['C2_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['±C1_f', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]],
['C2_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]]]],
['CC2_f', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['±C2_fe', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]],
['CD4_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['CD4_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['±C2_fk', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±C2_fc', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['D4_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['D4_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['CC4_fc', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]],
['±C4_fc', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]],
['DD8_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]],
['D8_f', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±D4_fk', [[[[0, -1, 0], [-1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['±D4_fe', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]],
['DD8_fe', [[[[0, -1, 0], [-1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]],
['±D8_f', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]],
['C2_ek', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['±C1_e', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['±C2_ef', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]],
['±C2_ec', [[[[-1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]]]],
['±C2_ek', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['D4_ec', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['CD4_eko', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['C4_ek', [[[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]],
['CD4_ekd', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['D4_ef', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['CC4_ek', [[[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]],
['±C4_ek', [[[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['DD8_ec', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]],
['D8_e', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]],
['±D4_ec', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]],
['±D4_ef', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['CD8_ek', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]],
['DD8_ef', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]],
['±D8_e', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['±C1_k', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]],
['D4_ke', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]]]],
['CC4_ke', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]],
['±C2_kf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]],
['±C2_ke', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]]]],
['D4_kf', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]]]],
['D6_k', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]],
['±C3_k', [[[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [1, 1, 1]]]],
['±D4_ke', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [1, 0, 0]]]],
['±C4_ke', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]],
['±D4_kf', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]]]],
['DD8_kf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]],
['DD8_ke', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]],
['D8_k', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]]]],
['±D6_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [1, 1, 1]]]],
['T12_k', [[[[0, 0, -1], [-1, 0, 0], [0, 1, 0]], [1, 1, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]],
['±D8_k', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]],
['±T12_k', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]], [[[0, 1, 0], [0, 0, 1], [-1, 0, 0]], [0, 0, 1]]]],
['TO24_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [1, 1, 0]]]],
['O24_k', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]],
['±O24_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]]
]</nowiki>
</div>
</div>

And here we present the same generators in a more human-readable format, specifically where they send the vector [x,y,z].
<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view the generators data.
<div class="mw-collapsible-content">
<nowiki>
[
['C1', []],
['C2_cf', [[-x, y, -z]]],
['C2_ce', [[-y, -x, -z]]],
['CC2_/', [[y, x, z]]],
['±C1_c', [[-x, -y, -z]]],
['CC2_c', [[-x, y, z]]],
['C3', [[y, z, x]]],
['D4_cf', [[-x, -y, z], [-x, y, -z]]],
['C4_cf', [[-y, x, z]]],
['CC4_cf', [[y, -x, -z]]],
['CD4_cfd', [[y, x, z], [-x, -y, z]]],
['CD4_ce', [[y, x, z], [x, y, -z]]],
['±C2_ce', [[-x, -y, -z], [y, x, z]]],
['±C2_cf', [[x, -y, z], [-x, y, -z]]],
['CD4_cfo', [[x, y, -z], [-x, y, -z]]],
['D4_ce', [[-y, -x, -z], [-x, -y, z]]],
['CD6', [[y, x, z], [y, z, x]]],
['D6_c', [[-y, -x, -z], [y, z, x]]],
['±C3_c', [[-y, -z, -x]]],
['±D4_cf', [[-x, -y, z], [-x, y, -z], [-x, y, z]]],
['±C4_cf', [[-y, x, z], [-x, -y, -z]]],
['±D4_ce', [[x, y, -z], [y, x, z], [-x, -y, z]]],
['DD8_ce', [[x, -y, z], [y, -x, -z]]],
['DD8_cf', [[-x, y, -z], [y, -x, -z]]],
['CD8_cf', [[x, -y, z], [-y, x, z]]],
['D8_c', [[-x, y, -z], [-y, x, z]]],
['±D6_c', [[y, x, z], [-y, -z, -x]]],
['T12_c', [[-z, -x, y], [x, -y, -z]]],
['±D8_c', [[-x, y, -z], [-y, x, z], [-x, -y, -z]]],
['±T12_c', [[x, -y, -z], [y, z, -x]]],
['TO24_c', [[y, x, z], [-x, -z, y]]],
['O24_c', [[-y, -x, -z], [x, z, -y]]],
['±O24_c', [[y, x, z], [x, z, -y]]],
['C2_fe', [[-x, y, -z + 1]]],
['±C1_f', [[-x, -y, -z + 1]]],
['C2_fk', [[y, x, -z + 1]]],
['CC2_f', [[x, y, -z + 1]]],
['±C2_fe', [[-x, -y, -z + 1], [-x, y, z]]],
['CD4_fe', [[-x, y, -z + 1], [x, y, -z + 1]]],
['CD4_fk', [[y, x, -z + 1], [x, y, -z + 1]]],
['±C2_fk', [[-y, -x, -z + 1], [y, x, z]]],
['±C2_fc', [[x, y, -z + 1], [-x, -y, z]]],
['D4_fk', [[y, x, -z + 1], [-x, -y, z]]],
['D4_fe', [[-x, y, -z + 1], [-x, -y, z]]],
['CC4_fc', [[y, -x, -z + 1]]],
['±C4_fc', [[-y, x, z], [-x, -y, -z + 1]]],
['DD8_fk', [[y, x, -z + 1], [y, -x, -z + 1]]],
['D8_f', [[y, x, -z + 1], [-y, x, z]]],
['±D4_fk', [[-y, -x, z], [y, x, z], [x, y, -z + 1]]],
['±D4_fe', [[x, y, -z + 1], [-x, y, z], [-x, -y, z]]],
['DD8_fe', [[-y, -x, z], [y, -x, -z + 1]]],
['±D8_f', [[y, x, -z + 1], [-y, x, z], [-x, -y, -z + 1]]],
['C2_ek', [[x, -y + 1, -z + 1]]],
['±C1_e', [[-x, -y + 1, -z + 1]]],
['±C2_ef', [[-x, -y + 1, -z + 1], [x, y, -z + 1]]],
['±C2_ec', [[-x, -z + 1, -y + 1], [x, z, y]]],
['±C2_ek', [[-x, y, z], [x, -y + 1, -z + 1]]],
['D4_ec', [[-x, z, y], [x, -y + 1, -z + 1]]],
['CD4_eko', [[x, -y + 1, z], [x, -y + 1, -z + 1]]],
['C4_ek', [[x, z, -y + 1]]],
['CD4_ekd', [[x, -z + 1, -y + 1], [x, -y + 1, -z + 1]]],
['D4_ef', [[-x, y, -z + 1], [x, -y + 1, -z + 1]]],
['CC4_ek', [[-x, -z + 1, y]]],
['±C4_ek', [[x, z, -y + 1], [-x, -y + 1, -z + 1]]],
['DD8_ec', [[-x, z, y], [-x, -z + 1, y]]],
['D8_e', [[-x, z, y], [x, z, -y + 1]]],
['±D4_ec', [[x, -z + 1, -y + 1], [x, z, y], [-x, y, z]]],
['±D4_ef', [[-x, y, z], [x, y, -z + 1], [x, -y + 1, -z + 1]]],
['CD8_ek', [[x, -z + 1, -y + 1], [x, z, -y + 1]]],
['DD8_ef', [[x, -z + 1, -y + 1], [-x, -z + 1, y]]],
['±D8_e', [[-x, z, y], [x, z, -y + 1], [-x, -y + 1, -z + 1]]],
['±C1_k', [[-x + 1, -y + 1, -z + 1]]],
['D4_ke', [[-x + 1, -y + 1, z], [-x + 1, y, -z + 1]]],
['CC4_ke', [[y, -x + 1, -z + 1]]],
['±C2_kf', [[-x + 1, -y + 1, -z + 1], [y, x, z]]],
['±C2_ke', [[x, -y + 1, z], [-x + 1, y, -z + 1]]],
['D4_kf', [[-y + 1, -x + 1, -z + 1], [-x + 1, -y + 1, z]]],
['D6_k', [[-y + 1, -x + 1, -z + 1], [y, z, x]]],
['±C3_k', [[-y + 1, -z + 1, -x + 1]]],
['±D4_ke', [[-x + 1, -y + 1, z], [-x + 1, y, -z + 1], [-x + 1, y, z]]],
['±C4_ke', [[-y + 1, x, z], [-x + 1, -y + 1, -z + 1]]],
['±D4_kf', [[x, y, -z + 1], [y, x, z], [-x + 1, -y + 1, z]]],
['DD8_kf', [[x, -y + 1, z], [y, -x + 1, -z + 1]]],
['DD8_ke', [[-x + 1, y, -z + 1], [y, -x + 1, -z + 1]]],
['D8_k', [[-x + 1, y, -z + 1], [-y + 1, x, z]]],
['±D6_k', [[y, x, z], [-y + 1, -z + 1, -x + 1]]],
['T12_k', [[-z + 1, -x + 1, y], [x, -y + 1, -z + 1]]],
['±D8_k', [[-x + 1, y, -z + 1], [-y + 1, x, z], [-x + 1, -y + 1, -z + 1]]],
['±T12_k', [[x, -y + 1, -z + 1], [y, z, -x + 1]]],
['TO24_k', [[y, x, z], [-x + 1, -z + 1, y]]],
['O24_k', [[-y + 1, -x + 1, -z + 1], [x, z, -y + 1]]],
['±O24_k', [[y, x, z], [x, z, -y + 1]]]
]</nowiki>
</div>
</div>

== Objects ==
Below is a list of what are, I believe, minimal objects with the corresponding symmetry group. The objects are given as lists of points.

<div class="mw-collapsible mw-collapsed">
Click on "Expand" to the right to view the points.
<div class="mw-collapsible-content">
<nowiki>
[
['C1', [[0, -1, -1], [0, 0, 0], [1, 0, 1], [1, 1, 1]]],
['C2_cf', [[-1, 0, -1], [0, 0, -1], [0, 0, 1], [0, 1, 0], [1, 0, 1]]],
['C2_ce', [[0, -1, -1], [0, 0, 0], [1, 0, 1]]],
['CC2_/', [[-1, -1, -1], [0, 0, 0], [1, 1, 0]]],
['±C1_c', [[-1, -1, -2], [-1, 0, -1], [0, 0, 0], [1, 0, 1], [1, 1, 2]]],
['CC2_c', [[0, -1, 0], [0, 0, 0], [0, 1, 1]]],
['C3', [[-1, 0, 1], [0, 0, 1], [0, 1, -1], [0, 1, 0], [1, -1, 0], [1, 0, 0]]],
['D4_cf', [[-1, -1, 1], [-1, 0, 0], [-1, 1, -1], [0, -2, 0], [0, 2, 0], [1, -1, -1], [1, 0, 0], [1, 1, 1]]],
['C4_cf', [[-2, -1, 1], [-1, 0, 0], [-1, 2, 1], [0, -1, 0], [0, 1, 0], [1, -2, 1], [1, 0, 0], [2, 1, 1]]],
['CC4_cf', [[-2, -1, 1], [-1, 0, 0], [-1, 2, -1], [0, -1, 0], [0, 1, 0], [1, -2, -1], [1, 0, 0], [2, 1, 1]]],
['CD4_cfd', [[-1, -1, 1], [0, 0, 0], [1, 1, 1]]],
['CD4_ce', [[0, 0, 0], [0, 1, 0], [1, 0, 0]]],
['±C2_ce', [[-1, -1, -1], [-1, -1, 0], [0, 0, 0], [1, 1, 0], [1, 1, 1]]],
['±C2_cf', [[-1, 0, -1], [-1, 0, 0], [0, 0, 0], [1, 0, 0], [1, 0, 1]]],
['CD4_cfo', [[-1, 1, 0], [0, 0, 0], [1, 1, 0]]],
['D4_ce', [[-2, -1, 1], [-1, -2, -1], [-1, -1, 0], [0, 0, 0], [1, 1, 0], [1, 2, -1], [2, 1, 1]]],
['CD6', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0]]],
['D6_c', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [1, -1, 0]]],
['±C3_c', [[-2, 0, -1], [-1, -2, 0], [-1, -1, -1], [0, -1, -2], [0, 0, 0], [0, 1, 2], [1, 1, 1], [1, 2, 0], [2, 0, 1]]],
['±D4_cf', [[-1, -1, 0], [-1, 0, 0], [-1, 1, 0], [0, 0, 0], [1, -1, 0], [1, 0, 0], [1, 1, 0]]],
['±C4_cf', [[-2, -1, 0], [-1, 0, 0], [-1, 2, 0], [0, -1, 0], [0, 1, 0], [1, -2, 0], [1, 0, 0], [2, 1, 0]]],
['±D4_ce', [[-1, -1, 0], [0, 0, 0], [1, 1, 0]]],
['DD8_ce', [[-1, 0, 1], [0, -1, -1], [0, 0, 0], [0, 1, -1], [1, 0, 1]]],
['DD8_cf', [[-1, -1, 1], [-1, 1, -1], [0, 0, -1], [0, 0, 0], [0, 0, 1], [1, -1, -1], [1, 1, 1]]],
['CD8_cf', [[-1, 0, 1], [0, -1, 1], [0, 0, 0], [0, 1, 1], [1, 0, 1]]],
['D8_c', [[-2, -1, 1], [-2, 1, -1], [-1, -2, -1], [-1, 0, 0], [-1, 2, 1], [0, -1, 0], [0, 1, 0], [1, -2, 1], [1, 0, 0], [1, 2, -1], [2, -1, -1], [2, 1, 1]]],
['±D6_c', [[-1, -1, -1], [0, 0, 0], [1, 1, 1]]],
['T12_c', [[-2, 0, -1], [-2, 0, 1], [-1, -2, 0], [-1, -1, 1], [-1, 1, -1], [-1, 2, 0], [0, -1, -2], [0, -1, 2], [0, 0, 0], [0, 1, -2], [0, 1, 2], [1, -2, 0], [1, -1, -1], [1, 1, 1], [1, 2, 0], [2, 0, -1], [2, 0, 1]]],
['±D8_c', [[0, 0, -1], [0, 0, 0], [0, 0, 1]]],
['±T12_c', [[-2, -1, 0], [-2, 1, 0], [-1, 0, -2], [-1, 0, 0], [-1, 0, 2], [0, -2, -1], [0, -2, 1], [0, -1, 0], [0, 0, -1], [0, 0, 1], [0, 1, 0], [0, 2, -1], [0, 2, 1], [1, 0, -2], [1, 0, 0], [1, 0, 2], [2, -1, 0], [2, 1, 0]]],
['TO24_c', [[-1, -1, 1], [-1, 1, -1], [0, 0, 0], [1, -1, -1], [1, 1, 1]]],
['O24_c', [[-3, -2, -1], [-3, -1, 2], [-3, 1, -2], [-3, 2, 1], [-2, -3, 1], [-2, -1, -3], [-2, -1, 0], [-2, 0, -1], [-2, 0, 1], [-2, 1, 0], [-2, 1, 3], [-2, 3, -1], [-1, -3, -2], [-1, -2, 0], [-1, -2, 3], [-1, 0, -2], [-1, 0, 2], [-1, 2, -3], [-1, 2, 0], [-1, 3, 2], [0, -2, -1], [0, -2, 1], [0, -1, -2], [0, -1, 2], [0, 1, -2], [0, 1, 2], [0, 2, -1], [0, 2, 1], [1, -3, 2], [1, -2, -3], [1, -2, 0], [1, 0, -2], [1, 0, 2], [1, 2, 0], [1, 2, 3], [1, 3, -2], [2, -3, -1], [2, -1, 0], [2, -1, 3], [2, 0, -1], [2, 0, 1], [2, 1, -3], [2, 1, 0], [2, 3, 1], [3, -2, 1], [3, -1, -2], [3, 1, 2], [3, 2, -1]]],
['±O24_c', [[0, 0, 0]]],
['C2_fe', [[-1, 1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0]]],
['±C1_f', [[-1, -1, 1], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 1, 0]]],
['C2_fk', [[0, 0, 0], [0, 0, 1], [0, 1, 1], [1, 0, 0]]],
['CC2_f', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]],
['±C2_fe', [[0, -1, 0], [0, 0, 0], [0, 0, 1], [0, 1, 1]]],
['CD4_fe', [[0, 0, 0], [0, 0, 1], [0, 1, -1], [0, 1, 2]]],
['CD4_fk', [[0, 0, 0], [0, 0, 1], [1, 1, -1], [1, 1, 2]]],
['±C2_fk', [[-1, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0]]],
['±C2_fc', [[-1, -1, 0], [-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]],
['D4_fk', [[-2, -1, 0], [-1, -2, 1], [-1, 0, 0], [0, -1, 1], [0, 1, 1], [1, 0, 0], [1, 2, 1], [2, 1, 0]]],
['D4_fe', [[-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 1, 0], [0, 0, 0], [0, 0, 1], [1, -1, 0], [1, 0, 0], [1, 0, 1], [1, 1, 1]]],
['CC4_fc', [[-1, -1, 2], [-1, 0, 0], [-1, 1, -1], [0, -1, 1], [0, 1, 1], [1, -1, -1], [1, 0, 0], [1, 1, 2]]],
['±C4_fc', [[-2, -1, 0], [-2, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 2, 0], [-1, 2, 1], [0, -1, 0], [0, -1, 1], [0, 1, 0], [0, 1, 1], [1, -2, 0], [1, -2, 1], [1, 0, 0], [1, 0, 1], [2, 1, 0], [2, 1, 1]]],
['DD8_fk', [[-1, 0, 0], [0, -1, 1], [0, 1, 1], [1, 0, 0]]],
['D8_f', [[-2, -1, 1], [-2, 1, 0], [-1, -2, 0], [-1, 0, 0], [-1, 0, 1], [-1, 2, 1], [0, -1, 0], [0, -1, 1], [0, 1, 0], [0, 1, 1], [1, -2, 1], [1, 0, 0], [1, 0, 1], [1, 2, 0], [2, -1, 0], [2, 1, 1]]],
['±D4_fk', [[-1, -1, 0], [-1, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]],
['±D4_fe', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 0, 1]]],
['DD8_fe', [[-1, -1, 0], [-1, 1, 1], [0, 0, 0], [0, 0, 1], [1, -1, 1], [1, 1, 0]]],
['±D8_f', [[0, 0, 0], [0, 0, 1]]],
['C2_ek', [[-1, 0, -1], [-1, 1, 2], [0, 0, 0], [0, 1, 1]]],
['±C1_e', [[-1, 2, 1], [0, 0, 0], [0, 1, 1], [1, -1, 0]]],
['±C2_ef', [[-1, 1, 0], [-1, 1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1]]],
['±C2_ec', [[-1, 1, 1], [0, 0, 0], [0, 1, 1], [1, 0, 0]]],
['±C2_ek', [[0, -1, 0], [0, 0, 0], [0, 1, 1], [0, 2, 1]]],
['D4_ec', [[-1, 0, -1], [-1, 1, 2], [0, 0, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]],
['CD4_eko', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 2, 0], [1, 2, 1]]],
['C4_ek', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 0, 2], [1, 1, -1], [1, 2, 1]]],
['CD4_ekd', [[0, 0, 0], [0, 1, 1], [1, -1, -1], [1, 2, 2]]],
['D4_ef', [[-1, -1, 1], [-1, 2, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]],
['CC4_ek', [[-1, 0, 2], [-1, 1, -1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]],
['±C4_ek', [[0, -1, 0], [0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, -1], [0, 1, 0], [0, 1, 1], [0, 2, 1]]],
['DD8_ec', [[-1, 0, -1], [-1, 0, 2], [-1, 1, -1], [-1, 1, 2], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 2, 0], [1, 2, 1]]],
['D8_e', [[-1, -1, 1], [-1, 0, -1], [-1, 1, 2], [-1, 2, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 0, 2], [1, 1, -1], [1, 2, 1]]],
['±D4_ec', [[0, 0, 0], [0, 1, 1]]],
['±D4_ef', [[0, -1, 0], [0, -1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, 0], [0, 2, 1]]],
['CD8_ek', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 0, -1], [1, 0, 2], [1, 1, -1], [1, 1, 2], [1, 2, 0], [1, 2, 1]]],
['DD8_ef', [[-1, 0, 1], [-1, 1, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 1, 1]]],
['±D8_e', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1]]],
['±C1_k', [[-1, 0, 0], [0, 1, 0], [1, 0, 1], [2, 1, 1]]],
['D4_ke', [[-1, -1, 0], [-1, 0, 0], [-1, 1, 1], [-1, 2, 1], [0, -2, 0], [0, 3, 1], [1, -2, 1], [1, 3, 0], [2, -1, 1], [2, 0, 1], [2, 1, 0], [2, 2, 0]]],
['CC4_ke', [[-1, 0, -1], [0, 0, 0], [0, 1, 1], [0, 2, 2], [1, -1, 2], [1, 0, 1], [1, 1, 0], [2, 1, -1]]],
['±C2_kf', [[0, 0, -1], [0, 0, 0], [1, 1, 1], [1, 1, 2]]],
['±C2_ke', [[-1, 0, 0], [-1, 1, 0], [0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1], [2, 0, 1], [2, 1, 1]]],
['D4_kf', [[-1, 0, 0], [0, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1], [1, 2, 1], [2, 1, 0]]],
['D6_k', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [0, 1, 2], [1, -1, 0], [1, 1, 1], [1, 2, 0], [2, 0, 1]]],
['±C3_k', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [0, 2, 1], [1, -1, 0], [1, 0, 2], [1, 1, 1], [2, 1, 0]]],
['±D4_ke', [[-1, -1, 0], [-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 1, 0], [-1, 1, 1], [-1, 2, 0], [-1, 2, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [2, -1, 0], [2, -1, 1], [2, 0, 0], [2, 0, 1], [2, 1, 0], [2, 1, 1], [2, 2, 0], [2, 2, 1]]],
['±C4_ke', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, 0], [0, 2, 1], [1, -1, 0], [1, -1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [2, 1, 0], [2, 1, 1]]],
['±D4_kf', [[0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]],
['DD8_kf', [[-1, 0, 0], [-1, 1, 0], [0, -1, 1], [0, 2, 1], [1, -1, 1], [1, 2, 1], [2, 0, 0], [2, 1, 0]]],
['DD8_ke', [[0, 0, -1], [0, 0, 0], [0, 1, 1], [0, 1, 2], [1, 0, 1], [1, 0, 2], [1, 1, -1], [1, 1, 0]]],
['D8_k', [[-1, 0, -1], [-1, 1, 2], [0, -1, 2], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, -1], [1, -1, -1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [1, 2, 2], [2, 0, 2], [2, 1, -1]]],
['±D6_k', [[0, 0, 0], [1, 1, 1]]],
['T12_k', [[-2, 0, 2], [-2, 1, -1], [-1, -2, 1], [-1, 0, 1], [-1, 1, 0], [-1, 3, 0], [0, -1, 1], [0, -1, 3], [0, 0, 2], [0, 1, -1], [0, 2, -2], [0, 2, 0], [1, -1, -2], [1, -1, 0], [1, 0, -1], [1, 1, 2], [1, 2, 1], [1, 2, 3], [2, -2, 0], [2, 0, 0], [2, 1, 1], [2, 3, 1], [3, 0, -1], [3, 1, 2]]],
['±D8_k', [[0, 0, -1], [0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, -1], [0, 1, 0], [0, 1, 1], [0, 1, 2], [1, 0, -1], [1, 0, 0], [1, 0, 1], [1, 0, 2], [1, 1, -1], [1, 1, 0], [1, 1, 1], [1, 1, 2]]],
['±T12_k', [[-2, -1, 0], [-2, -1, 1], [-2, 2, 0], [-2, 2, 1], [-1, -1, -1], [-1, -1, 2], [-1, 0, -2], [-1, 0, 3], [-1, 1, -2], [-1, 1, 3], [-1, 2, -1], [-1, 2, 2], [0, -2, -1], [0, -2, 2], [0, 3, -1], [0, 3, 2], [1, -2, -1], [1, -2, 2], [1, 3, -1], [1, 3, 2], [2, -1, -1], [2, -1, 2], [2, 0, -2], [2, 0, 3], [2, 1, -2], [2, 1, 3], [2, 2, -1], [2, 2, 2], [3, -1, 0], [3, -1, 1], [3, 2, 0], [3, 2, 1]]],
['TO24_k', [[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]],
['O24_k', [[-2, -1, 1], [-2, 0, -1], [-2, 1, 2], [-2, 2, 0], [-1, -2, 0], [-1, -1, -1], [-1, -1, 2], [-1, 0, 3], [-1, 1, -2], [-1, 2, -1], [-1, 2, 2], [-1, 3, 1], [0, -2, 2], [0, -1, -2], [0, 2, 3], [0, 3, -1], [1, -2, -1], [1, -1, 3], [1, 2, -2], [1, 3, 2], [2, -2, 1], [2, -1, -1], [2, -1, 2], [2, 0, -2], [2, 1, 3], [2, 2, -1], [2, 2, 2], [2, 3, 0], [3, -1, 0], [3, 0, 2], [3, 1, -1], [3, 2, 1]]],
['±O24_k', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]]
]</nowiki>
</div>
</div>

== Kinetic symmetries ==
In addition to the 92 above static symmetries there are also 245 [[Kinetic symmetry|kinetic symmetries]] with [[mod]] unequal to their [[period]] (220 with period/mod=2, 7 with period/mod=3, 14 with period/mod=4, and 4 with period/mod=6).

"Composite symmetry" refers to the resulting symmetry of the pattern created from each of the oscillator's phases.

{| class="wikitable"
! Static symmetry
! Composite symmetry
! period/mod
! Description
! [[Gutteroid]]s
|-
! C1
! C2_cf
| 2
|
|
|-
! C1
! C2_ce
| 2
|
|
|-
! C1
! CC2_/
| 2
|
|
|-
! C1
! ±C1_c
| 2
|
|
|-
! C1
! CC2_c
| 2
|
|
|-
! C2_cf
! D4_cf
| 2
|
|
|-
! C2_cf
! C4_cf
| 2
|
|
|-
! C2_cf
! CC4_cf
| 2
|
|
|-
! C2_cf
! CD4_cfd
| 2
|
|
|-
! CC2_/
! CD4_cfd
| 2
|
|
|-
! C2_ce
! CD4_ce
| 2
|
|
|-
! CC2_/
! CD4_ce
| 2
|
|
|-
! CC2_c
! CD4_ce
| 2
|
|
|-
! C2_ce
! ±C2_ce
| 2
|
|
|-
! CC2_/
! ±C2_ce
| 2
|
|
|-
! ±C1_c
! ±C2_ce
| 2
|
|
|-
! C2_cf
! ±C2_cf
| 2
|
|
|-
! ±C1_c
! ±C2_cf
| 2
|
|
|-
! CC2_c
! ±C2_cf
| 2
|
|
|-
! C2_cf
! CD4_cfo
| 2
|
|
|-
! CC2_c
! CD4_cfo
| 2
|
|
|-
! C2_cf
! D4_ce
| 2
|
|
|-
! C2_ce
! D4_ce
| 2
|
|
|-
! C3
! CD6
| 2
|
|
|-
! C3
! D6_c
| 2
|
|
|-
! C3
! ±C3_c
| 2
|
|
|-
! D4_cf
! ±D4_cf
| 2
|
|
|-
! ±C2_cf
! ±D4_cf
| 2
|
|
|-
! CD4_cfo
! ±D4_cf
| 2
|
|
|-
! C4_cf
! ±C4_cf
| 2
|
|
|-
! CC4_cf
! ±C4_cf
| 2
|
|
|-
! ±C2_cf
! ±C4_cf
| 2
|
|
|-
! CD4_cfd
! ±D4_ce
| 2
|
|
|-
! CD4_ce
! ±D4_ce
| 2
|
|
|-
! ±C2_ce
! ±D4_ce
| 2
|
|
|-
! ±C2_cf
! ±D4_ce
| 2
|
|
|-
! D4_ce
! ±D4_ce
| 2
|
|
|-
! CC4_cf
! DD8_ce
| 2
|
|
|-
! CD4_cfo
! DD8_ce
| 2
|
|
|-
! D4_ce
! DD8_ce
| 2
|
|
|-
! D4_cf
! DD8_cf
| 2
|
|
|-
! CC4_cf
! DD8_cf
| 2
|
|
|-
! CD4_cfd
! DD8_cf
| 2
|
|
|-
! C4_cf
! CD8_cf
| 2
|
|
|-
! CD4_cfd
! CD8_cf
| 2
|
|
|-
! CD4_cfo
! CD8_cf
| 2
|
|
|-
! D4_cf
! D8_c
| 2
|
|
|-
! C4_cf
! D8_c
| 2
|
|
|-
! D4_ce
! D8_c
| 2
|
|
|-
! CD6
! ±D6_c
| 2
|
|
|-
! D6_c
! ±D6_c
| 2
|
|
|-
! ±C3_c
! ±D6_c
| 2
|
|
|-
! ±D4_cf
! ±D8_c
| 2
|
|
|-
! ±C4_cf
! ±D8_c
| 2
|
|
|-
! ±D4_ce
! ±D8_c
| 2
|
|
|-
! DD8_ce
! ±D8_c
| 2
|
|
|-
! DD8_cf
! ±D8_c
| 2
|
|
|-
! CD8_cf
! ±D8_c
| 2
|
|
|-
! D8_c
! ±D8_c
| 2
|
|
|-
! T12_c
! ±T12_c
| 2
|
|
|-
! T12_c
! TO24_c
| 2
|
|
|-
! T12_c
! O24_c
| 2
|
|
|-
! ±T12_c
! ±O24_c
| 2
|
|
|-
! TO24_c
! ±O24_c
| 2
|
|
|-
! O24_c
! ±O24_c
| 2
|
|
|-
! C1
! C2_fe
| 2
|
|
|-
! C1
! ±C1_f
| 2
|
|
|-
! C1
! C2_fk
| 2
|
|
|-
! C1
! CC2_f
| 2
|
|
|-
! C2_fe
! ±C2_fe
| 2
|
|
|-
! CC2_c
! ±C2_fe
| 2
|
|
|-
! ±C1_f
! ±C2_fe
| 2
|
|
|-
! C2_fe
! CD4_fe
| 2
|
|
|-
! CC2_c
! CD4_fe
| 2
|
|
|-
! CC2_f
! CD4_fe
| 2
|
|
|-
! CC2_/
! CD4_fk
| 2
|
|
|-
! C2_fk
! CD4_fk
| 2
|
|
|-
! CC2_f
! CD4_fk
| 2
|
|
|-
! CC2_/
! ±C2_fk
| 2
|
|
|-
! ±C1_f
! ±C2_fk
| 2
|
|
|-
! C2_fk
! ±C2_fk
| 2
|
|
|-
! C2_cf
! ±C2_fc
| 2
|
|
|-
! ±C1_f
! ±C2_fc
| 2
|
|
|-
! CC2_f
! ±C2_fc
| 2
|
|
|-
! C2_cf
! D4_fk
| 2
|
|
|-
! C2_fk
! D4_fk
| 2
|
|
|-
! C2_cf
! D4_fe
| 2
|
|
|-
! C2_fe
! D4_fe
| 2
|
|
|-
! C2_cf
! CC4_fc
| 2
|
|
|-
! ±C2_fc
! ±C4_fc
| 2
|
|
|-
! C4_cf
! ±C4_fc
| 2
|
|
|-
! CC4_fc
! ±C4_fc
| 2
|
|
|-
! D4_fk
! DD8_fk
| 2
|
|
|-
! CD4_cfo
! DD8_fk
| 2
|
|
|-
! CC4_fc
! DD8_fk
| 2
|
|
|-
! D4_fk
! D8_f
| 2
|
|
|-
! C4_cf
! D8_f
| 2
|
|
|-
! D4_fe
! D8_f
| 2
|
|
|-
! CD4_fk
! ±D4_fk
| 2
|
|
|-
! ±C2_fk
! ±D4_fk
| 2
|
|
|-
! ±C2_fc
! ±D4_fk
| 2
|
|
|-
! D4_fk
! ±D4_fk
| 2
|
|
|-
! CD4_cfd
! ±D4_fk
| 2
|
|
|-
! ±C2_fe
! ±D4_fe
| 2
|
|
|-
! CD4_fe
! ±D4_fe
| 2
|
|
|-
! ±C2_fc
! ±D4_fe
| 2
|
|
|-
! CD4_cfo
! ±D4_fe
| 2
|
|
|-
! D4_fe
! ±D4_fe
| 2
|
|
|-
! CD4_cfd
! DD8_fe
| 2
|
|
|-
! D4_fe
! DD8_fe
| 2
|
|
|-
! CC4_fc
! DD8_fe
| 2
|
|
|-
! ±C4_fc
! ±D8_f
| 2
|
|
|-
! DD8_fk
! ±D8_f
| 2
|
|
|-
! D8_f
! ±D8_f
| 2
|
|
|-
! ±D4_fk
! ±D8_f
| 2
|
|
|-
! ±D4_fe
! ±D8_f
| 2
|
|
|-
! CD8_cf
! ±D8_f
| 2
|
|
|-
! DD8_fe
! ±D8_f
| 2
|
|
|-
! C1
! C2_ek
| 2
|
|
|-
! C1
! ±C1_e
| 2
|
|
|-
! C2_fe
! ±C2_ef
| 2
|
|
|-
! CC2_f
! ±C2_ef
| 2
|
|
|-
! ±C1_e
! ±C2_ef
| 2
|
|
|-
! CC2_/
! ±C2_ec
| 2
|
|
|-
! ±C1_e
! ±C2_ec
| 2
|
|
|-
! C2_ce
! ±C2_ec
| 2
|
|
|-
! C2_ek
! ±C2_ek
| 2
|
|
|-
! ±C1_e
! ±C2_ek
| 2
|
|
|-
! CC2_c
! ±C2_ek
| 2
|
|
|-
! C2_ek
! D4_ec
| 2
|
|
|-
! C2_ce
! D4_ec
| 2
|
|
|-
! C2_ek
! CD4_eko
| 2
|
|
|-
! CC2_f
! CD4_eko
| 2
|
|
|-
! C2_ek
! C4_ek
| 2
|
|
|-
! CC2_/
! CD4_ekd
| 2
|
|
|-
! C2_ek
! CD4_ekd
| 2
|
|
|-
! C2_ek
! D4_ef
| 2
|
|
|-
! C2_fe
! D4_ef
| 2
|
|
|-
! C2_ek
! CC4_ek
| 2
|
|
|-
! ±C2_ek
! ±C4_ek
| 2
|
|
|-
! C4_ek
! ±C4_ek
| 2
|
|
|-
! CC4_ek
! ±C4_ek
| 2
|
|
|-
! D4_ec
! DD8_ec
| 2
|
|
|-
! CD4_eko
! DD8_ec
| 2
|
|
|-
! CC4_ek
! DD8_ec
| 2
|
|
|-
! D4_ec
! D8_e
| 2
|
|
|-
! C4_ek
! D8_e
| 2
|
|
|-
! D4_ef
! D8_e
| 2
|
|
|-
! CD4_ce
! ±D4_ec
| 2
|
|
|-
! ±C2_ec
! ±D4_ec
| 2
|
|
|-
! ±C2_ek
! ±D4_ec
| 2
|
|
|-
! D4_ec
! ±D4_ec
| 2
|
|
|-
! CD4_ekd
! ±D4_ec
| 2
|
|
|-
! ±C2_ef
! ±D4_ef
| 2
|
|
|-
! CD4_fe
! ±D4_ef
| 2
|
|
|-
! ±C2_ek
! ±D4_ef
| 2
|
|
|-
! CD4_eko
! ±D4_ef
| 2
|
|
|-
! D4_ef
! ±D4_ef
| 2
|
|
|-
! CD4_eko
! CD8_ek
| 2
|
|
|-
! C4_ek
! CD8_ek
| 2
|
|
|-
! CD4_ekd
! CD8_ek
| 2
|
|
|-
! CD4_ekd
! DD8_ef
| 2
|
|
|-
! D4_ef
! DD8_ef
| 2
|
|
|-
! CC4_ek
! DD8_ef
| 2
|
|
|-
! ±C4_ek
! ±D8_e
| 2
|
|
|-
! DD8_ec
! ±D8_e
| 2
|
|
|-
! D8_e
! ±D8_e
| 2
|
|
|-
! ±D4_ec
! ±D8_e
| 2
|
|
|-
! ±D4_ef
! ±D8_e
| 2
|
|
|-
! CD8_ek
! ±D8_e
| 2
|
|
|-
! DD8_ef
! ±D8_e
| 2
|
|
|-
! C1
! ±C1_k
| 2
|
|
|-
! C2_ek
! D4_ke
| 2
|
|
|-
! C2_ek
! CC4_ke
| 2
|
|
|-
! C2_fk
! ±C2_kf
| 2
|
|
|-
! CC2_/
! ±C2_kf
| 2
|
|
|-
! ±C1_k
! ±C2_kf
| 2
|
|
|-
! C2_ek
! ±C2_ke
| 2
|
|
|-
! ±C1_k
! ±C2_ke
| 2
|
|
|-
! CC2_f
! ±C2_ke
| 2
|
|
|-
! C2_ek
! D4_kf
| 2
|
|
|-
! C2_fk
! D4_kf
| 2
|
|
|-
! C3
! D6_k
| 2
|
|
|-
! C3
! ±C3_k
| 2
|
|
|-
! D4_ke
! ±D4_ke
| 2
|
|
|-
! ±C2_ke
! ±D4_ke
| 2
|
|
|-
! CD4_eko
! ±D4_ke
| 2
|
|
|-
! C4_ek
! ±C4_ke
| 2
|
|
|-
! CC4_ke
! ±C4_ke
| 2
|
|
|-
! ±C2_ke
! ±C4_ke
| 2
|
|
|-
! CD4_ekd
! ±D4_kf
| 2
|
|
|-
! CD4_fk
! ±D4_kf
| 2
|
|
|-
! ±C2_kf
! ±D4_kf
| 2
|
|
|-
! ±C2_ke
! ±D4_kf
| 2
|
|
|-
! D4_kf
! ±D4_kf
| 2
|
|
|-
! CC4_ke
! DD8_kf
| 2
|
|
|-
! CD4_eko
! DD8_kf
| 2
|
|
|-
! D4_kf
! DD8_kf
| 2
|
|
|-
! D4_ke
! DD8_ke
| 2
|
|
|-
! CC4_ke
! DD8_ke
| 2
|
|
|-
! CD4_ekd
! DD8_ke
| 2
|
|
|-
! D4_ke
! D8_k
| 2
|
|
|-
! C4_ek
! D8_k
| 2
|
|
|-
! D4_kf
! D8_k
| 2
|
|
|-
! CD6
! ±D6_k
| 2
|
|
|-
! D6_k
! ±D6_k
| 2
|
|
|-
! ±C3_k
! ±D6_k
| 2
|
|
|-
! ±D4_ke
! ±D8_k
| 2
|
|
|-
! ±C4_ke
! ±D8_k
| 2
|
|
|-
! ±D4_kf
! ±D8_k
| 2
|
|
|-
! DD8_kf
! ±D8_k
| 2
|
|
|-
! DD8_ke
! ±D8_k
| 2
|
|
|-
! CD8_ek
! ±D8_k
| 2
|
|
|-
! D8_k
! ±D8_k
| 2
|
|
|-
! T12_k
! ±T12_k
| 2
|
|
|-
! T12_k
! TO24_k
| 2
|
|
|-
! T12_k
! O24_k
| 2
|
|
|-
! ±T12_k
! ±O24_k
| 2
|
|
|-
! TO24_k
! ±O24_k
| 2
|
|
|-
! O24_k
! ±O24_k
| 2
|
|
|-
! C1
! C3
| 3
|
|
|-
! ±C1_c
! ±C3_c
| 3
|
|
|-
! D4_cf
! T12_c
| 3
|
|
|-
! ±D4_cf
! ±T12_c
| 3
|
|
|-
! ±C1_k
! ±C3_k
| 3
|
|
|-
! D4_ke
! T12_k
| 3
|
|
|-
! ±D4_ke
! ±T12_k
| 3
|
|
|-
! C1
! C4_cf
| 4
|
|
|-
! C1
! CC4_cf
| 4
|
|
|-
! ±C1_c
! ±C4_cf
| 4
|
|
|-
! CC2_c
! ±C4_cf
| 4
|
|
|-
! C1
! CC4_fc
| 4
|
|
|-
! ±C1_f
! ±C4_fc
| 4
|
|
|-
! CC2_f
! ±C4_fc
| 4
|
|
|-
! C1
! C4_ek
| 4
|
|
|-
! C1
! CC4_ek
| 4
|
|
|-
! ±C1_e
! ±C4_ek
| 4
|
|
|-
! CC2_c
! ±C4_ek
| 4
|
|
|-
! C1
! CC4_ke
| 4
|
|
|-
! ±C1_k
! ±C4_ke
| 4
|
|
|-
! CC2_f
! ±C4_ke
| 4
|
|
|-
! C1
! ±C3_c
| 6
|
|
|-
! D4_cf
! ±T12_c
| 6
|
|
|-
! C1
! ±C3_k
| 6
|
|
|-
! D4_ke
! ±T12_k
| 6
|
|
|}

==Spaceship symmetries==

===With period equal to mod===
There are 22 spaceship symmetries with period equal to the mod.
{| class="wikitable"
! Static equivalent
! Direction
! Description
|-
! C1
| any
| no symmetry
|-
! C2_cf
| orthogonal
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and faces)
|-
! C2_ce
| diagonal
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and edges)
|-
! CC2_/
| any in diagonal plane
| reflection (plane goes through cells, faces, edges, and vertices)
|-
! CC2_c
| any in orthogonal plane
| reflection (plane goes through cells, edges, and faces)
|-
! C3
| paradiagonal
| chiral triangular pyramidal symmetry, a.k.a., 120 degree rotation (line goes through vertices and cells)
|-
! C4_cf
| orthogonal
| chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through cells and faces)
|-
! CD4_cfd
| orthogonal
| digonal pyramidal symmetry (line goes through cells and faces), reflection planes have diagonal orientation
|-
! CD4_ce
| diagonal
| digonal pyramidal symmetry (line goes through cells and edges)
|-
! CD4_cfo
| orthogonal
| digonal pyramidal symmetry (line goes through cells and faces), reflection planes have orthogonal orientation
|-
! CD6
| paradiagonal
| triangular pyramidal symmetry (line goes through vertices and cells)
|-
! CD8_cf
| orthogonal
| square pyramidal symmetry (line goes through cells and faces)
|-
! C2_fe
| orthogonal
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and edges)
|-
! C2_fk
| diagonal
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and vertices)
|-
! CC2_f
| any in orthogonal plane
| reflection (plane goes through faces, edges, and vertices)
|-
! CD4_fe
| orthogonal
| digonal pyramidal symmetry (line goes through faces and edges)
|-
! CD4_fk
| diagonal
| digonal pyramidal symmetry (line goes through faces and vertices)
|-
! C2_ek
| orthogonal
| chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through edges and vertices)
|-
! CD4_eko
| orthogonal
| digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have orthogonal orientation
|-
! C4_ek
| orthogonal
| chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through edges and vertices)
|-
! CD4_ekd
| orthogonal
| digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have diagonal orientation
|-
! CD8_ek
| orthogonal
| square pyramidal symmetry (line goes through edges and vertices)
|}

===With period different from mod===
There are 43 spaceship symmetries with period different from mod.
{| class="wikitable"
! Static symmetry
! Composite symmetry
! period/mod
! direction
! Description
|-
! C1
! C2_cf
| 2
| orthogonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through cells and faces)
|-
! C1
! C2_fe
| 2
| orthogonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through faces and edges)
|-
! C1
! C2_ek
| 2
| orthogonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through edges and vertices)
|-
! C1
! C2_ce
| 2
| diagonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through cells and edges)
|-
! C1
! C2_fk
| 2
| diagonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through faces and vertices)
|-
! C1
! N/A
| 2
| diagonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through edges)
|-
! C1
! N/A
| 2
| diagonal
| Pattern is asymmetric<br>180 degree skew-rotation (line goes through faces)
|-
! C1
! CC2_c
| 2
| any in orthogonal plane
| Pattern is asymmetric<br>skew-reflection (plane goes through cells, edges, and faces)
|-
! C1
! CC2_f
| 2
| any in orthogonal plane
| Pattern is asymmetric<br>skew-reflection (plane goes through faces, edges, and vertices)
|-
! C1
! CC2_/
| 2
| any in diagonal plane
| Pattern is asymmetric<br>skew-reflection (plane goes through cells, faces, edges, and vertices)
|-
! C1
! N/A
| 2
| any in diagonal plane
| Pattern is asymmetric<br>skew-reflection (plane goes through faces and edges)
|-
! C2_cf
! CD4_cfd
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through cells and faces)<br>skew-reflections through diagonal planes
|-
! C2_ek
! CD4_ekd
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through edges and vertices)<br>skew-reflections through diagonal planes
|-
! C2_cf
! CD4_cfo
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through cells and faces)<br>skew-reflections through orthogonal planes
|-
! C2_ek
! CD4_eko
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through edges and vertices)<br>skew-reflections through orthogonal planes
|-
! C2_cf
! C4_cf
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through cells and faces)<br>90 degree skew-rotation
|-
! C2_ek
! C4_ek
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through edges and vertices)<br>90 degree skew-rotation
|-
! C2_ce
! CD4_ce
| 2
| diagonal
| Pattern is symmetric under 180 degree rotation (line goes through cells and edges)<br>skew-reflections
|-
! C2_fk
! CD4_fk
| 2
| diagonal
| Pattern is symmetric under 180 degree rotation (line goes through faces and vertices)<br>skew-reflections
|-
! C2_fe
! CD4_fe
| 2
| orthogonal
| Pattern is symmetric under 180 degree rotation (line goes through faces and edges)<br>skew-reflections
|-
! CC2_/
! CD4_cfd
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices)<br>180 degree skew-rotation (line goes through cells and faces)
|-
! CC2_/
! CD4_ekd
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices)<br>180 degree skew-rotation (line goes through edges and vertices)
|-
! CC2_/
! CD4_ce
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices)<br>180 degree skew-rotation (line goes through cells and edges)
|-
! CC2_/
! CD4_fk
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices)<br>180 degree skew-rotation (line goes through faces and vertices)
|-
! CC2_c
! CD4_ce
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through cells, edges, and faces)<br>180 degree skew-rotation (line goes through cells and edges)
|-
! CC2_c
! N/A
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through cells, edges, and faces)<br>180 degree skew-rotation (line goes through edges)
|-
! CC2_f
! CD4_fk
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through faces, edges, and vertices)<br>180 degree skew-rotation (line goes through faces and vertices)
|-
! CC2_f
! N/A
| 2
| diagonal
| Pattern is symmetric under reflection (plane goes through faces, edges, and vertices)<br>180 degree skew-rotation (line goes through faces)
|-
! CC2_c
! CD4_cfo
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through cells, edges, and faces)<br>180 degree skew-rotation (line goes through cells and faces)
|-
! CC2_f
! CD4_eko
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through faces, edges, and vertices)<br>180 degree skew-rotation (line goes through edges and vertices)
|-
! CC2_c
! CD4_fe
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through cells, edges, and faces)<br>180 degree skew-rotation (line goes through faces and edges)
|-
! CC2_f
! CD4_fe
| 2
| orthogonal
| Pattern is symmetric under reflection (plane goes through faces, edges, and vertices)<br>180 degree skew-rotation (line goes through faces and edges)
|-
! C3
! CD6
| 2
| paradiagonal
| Pattern is symmetric under 120 degree rotation (line goes through cells and vertices)<br>skew-reflections
|-
! CD4_cfd
! CD8_cf
| 2
| orthogonal
| Pattern has digonal pyramidal symmetry (line goes through cells and faces), reflection planes have diagonal orientation<br>90 degree skew-rotation
|-
! CD4_ekd
! CD8_ek
| 2
| orthogonal
| Pattern has digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have diagonal orientation<br>90 degree skew-rotation
|-
! CD4_cfo
! CD8_cf
| 2
| orthogonal
| Pattern has digonal pyramidal symmetry (line goes through cells and faces), reflection planes have orthogonal orientation<br>90 degree skew-rotation
|-
! CD4_eko
! CD8_ek
| 2
| orthogonal
| Pattern has digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have orthogonal orientation<br>90 degree skew-rotation
|-
! C4_cf
! CD8_cf
| 2
| orthogonal
| Pattern is symmetric under 90 degree rotation (line goes through cells and faces)<br>skew-reflections
|-
! C4_ek
! CD8_ek
| 2
| orthogonal
| Pattern is symmetric under 90 degree rotation (line goes through edges and vertices)<br>skew-reflections
|-
! C1
! C3
| 3
| paradiagonal
| Pattern is asymmetric<br>120 degree skew-rotation (line goes through cells and vertices)
|-
! C1
! N/A
| 3
| paradiagonal
| Pattern is asymmetric<br>120 degree skew-rotation (line goes paradiagonally through edges and faces)
|-
! C1
! C4_cf
| 4
| orthogonal
| Pattern is asymmetric<br>90 degree skew-rotation (line goes through cells and faces)
|-
! C1
! C4_ek
| 4
| orthogonal
| Pattern is asymmetric<br>90 degree skew-rotation (line goes through edges and vertices)
|}
[[Category:Everything else]]

Silk moth

2026-02-26T19:05:50Z

Remu:

{{stub}}
{{Spaceship
|name = Silk moth
|pname = silkmoth
|c = 1190163
|bx = 199808
|by = 219828
|dir = Oblique
|slope = 15/11
|p = 107
|m = 107
|s = (15,11)c/107
|z = (15,11)c/107
|symmetry = n
|rulemin = B3/S23
|rulemax = B3/S23
|isorulemin = B3/S23
|isorulemax = B3/S23
|discoverer = Nico Brown
|discoveryear = 2026
|mc = true
}}

'''Silk moth''' is a [[(15,11)c/107]] [[spaceship]] constructed by [[Nico Brown]] and finalised on 11 January 2026.<ref name="post223859" /> It is an [[oblique spaceship]] based on a (15,11)c/107 [[I-heptomino]] climber. Its name comes from the fact that it resembles a lepidopteran wing. Another proposed name was ''sea spider''.

== Underlying Reaction ==

The underlying reaction consists of an [[I-heptomino]] climbing a stream of gliders, emitting two gliders and a block for every one glider it receives.

Like the [[(34,7)c/156 Herschel climber]], this reaction is not completely [[endemic]] to life and would theoretically also work for a completion in the rule [[B3/S234c]] (Conway++). The reason this ship ultimately fails in this rule is because the east-turning MWSS turner (in a ship travelling NNE) triggers S4c. <ref name="post223904" />

== References ==
<references>
<ref name="post223859">{{LinkForumThread
|format = ref
|p = 223859
|title = Re: (15,11)c/107 caterpillar thread
|author = Nico Brown
|date = January 11, 2026
}}</ref>
<ref name="post223904">{{LinkForumThread
|format = ref
|p = 223904
|title = Re: (15,11)c/107 caterpillar thread
|author = LaundryPizza03
|date = January 12, 2026
}}</ref>
</references>
[[Category:Engineered spaceships]] [[Category:Macro-spaceships]]