LifeWiki - User contributions [en]

Cyclic

2024-11-14T23:43:45Z

JsfasdF248: grammar

{{Oscillator
|name = Cyclic
|pname = cyclic
|f1 = Billiard table
|c = 32
|bx = 10
|by = 10
|fc = 42.5
|p = 2
|m = 2
|h = 8
|v = 0.22
|symmetry = xk
|rotor = Diagonal on-off
|discoverer = Ian Osgood
|discoveryear = 2006
|rulemin = B3/S23
|rulemax = B378/S01235678
|rulespecial = [[Conway's Game of Life|Conway Life]]
|isorulemin = B3jn/S2ck3jknqy
|isorulemax = B2cik34-ejwz5-kq6-n78/S01234-ky5678
|synthesis = 9
|synthesisRLE = true
|animated = true
|plaintext = true
|rle = true
|viewerconfig = #C [[ LOOP 2 THUMBSIZE 2 HEIGHT 480 GPS 2 ]]
|apgcode = xp2_g8861dmo8gz123dmgc221
|pentadecathlonid = 32P2.15192
}}
'''Cyclic''' is a {{period|2}} [[oscillator]] that resembles both a [[bakery]] and the rotor of the [[great on-off]] and [[eater plug]]. It was reported by Ian Osgood in {{year|2006}} and recognized as a new [[casing]] for a previously known oscillator, {{LinkCatagolue|xp2_0g842dlq4oz34blm8421|patternname=32P2.15181|style=raw}}.<ref>{{LinkGoLNews|2006/03/new_p2_oscillators.html|title=New P2 Oscillators|date=March 1, 2006|author=Heinrich Koenig‏‎}}</ref>

One predecessor consists entirely of a bakery plus two diagonally adjacent cells added in the middle, as shown below. Due to the simplicity of this predecessor, the oscillator is sometimes referred to as the "most rediscovered pattern" in Conway's Game of Life.<ref name="post121959" />

Another predecessor consists of a [[lake 2]] with a [[duoplet]] in the middle, which is also frequently rediscovered.

{{gallery top}}
{{gallery item|{{EmbedViewer
|pname = cyclicpredecessor
|position = center
|caption = Cyclic predecessor
|style = width:300px;
}}}}

{{gallery item|{{EmbedViewer
|rle = x = 10, y = 10, rule = B3/S23
4b2o$3bo2bo$3bo2bo$b2o4b2o$o3bo4bo$o4bo3bo$b2o4b2o$3bo2bo$3bo2bo$4b2o!
|position = center
|caption = Another cyclic predecessor
|style = width:300px;
}}}}
{{gallery bottom}}

A {{gliders|10}} [[synthesis]] for this oscillator was found by [[Goldtiger997]] on July 6, 2016.<ref name="post32885" />

== See also ==
* [[A for all]]
* [[Four boats]]
* [[Washing machine]]

== References ==
<references>
<ref name="post121959">{{LinkForumThread
|format = ref
|title = Re: Thread For Your Useless Discoveries
|p = 121959
|author = bubblegum
|date = February 6, 2021
}}</ref>
<ref name="post32885">{{LinkForumThread
|format = ref
|p = 32885
|title = Re: Synthesising Oscillators
|author = Goldtiger997
|date = July 6, 2016
}}</ref>
</references>

== External links ==
* {{LinkCatagolue|xp2_g8861dmo8gz123dmgc221}}
* {{LinkPentadecathlonObject|32P2.15192}}

Density

2022-09-07T15:12:03Z

JsfasdF248: /* Still Lifes */ sortable

{{Glossary}}
The '''density''' of a pattern is the proportion of live [[cell]]s within its [[bounding box]].

==Still Lifes==
The general problem of finding the maximum density of an nxn square [[pseudo]] [[still life]] was solved for all n in 2012.<ref name=ref1>{{cite web|url=https://people.eng.unimelb.edu.au/pstuckey/papers/still-life.pdf | title=A Complete Solution to the Maximum Density Still Life Problem | publisher=Geoffrey Chu, Peter J. Stuckey, NICTA Victoria Laboratory, Department of Computing and Information Systems, University of Melbourne, Australia | date=November 8, 2013}}</ref> The maximum densities of square still lifes up to 20×20 were known ten years earlier.<ref>{{cite web|url=https://web.archive.org/web/20130731001049/http://www.ai.sri.com/~nysmith/life |title=Maximum Density Still Life |publisher=Neil Yorke-Smith|date=August 4, 2003}}</ref>

The following table contains maximum density still life information for side lengths through 25.

{|class="wikitable sortable" style="text-align:center"
|-
! Side length !! Max. density !! Max. population&dagger; (example) !! Example pattern(s) !! Total patterns
|-
| 1 || 0 || 0 || || 0
|-
| 2 || 1 || 4 [[block]] || {{EmbedViewer|pname=block|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 1
|-
| 3 || 0.6667 || 6 [[ship]] || {{EmbedViewer|pname=ship|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 1
|-
| 4 || 0.5 || 8 [[pond]] [[long ship]] || {{EmbedViewer|pname=pond|viewerconfig= #C [[ THUMBSIZE 3 ]]}} {{EmbedViewer|pname=longship|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 2
|-
| 5 || 0.64 || 16 [https://catagolue.hatsya.com/object/xs16_rr0rr/b3s23 four blocks] || {{EmbedViewer|pname=4blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 1
|-
| 6 || 0.5 || 18 two [[long shillelagh]]s [https://catagolue.hatsya.com/object/xs18_j5mgdbz1101/b3s23 trans-snake-tie with ship] [https://catagolue.hatsya.com/object/xs18_gbr0rrz11/b3s23 three blocks with ship] || {{EmbedViewer|pname=twolongshillelaghs|viewerconfig= #C [[ THUMBSIZE 3 ]]}} {{EmbedViewer|pname=transsnaketiewithship|viewerconfig= #C [[ THUMBSIZE 3 ]]}} {{EmbedViewer|pname=3blockswithship|viewerconfig= #C [[ THUMBSIZE 3 ]]}} {{EmbedViewer|pname=6x6maxdensitystilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 9
|-
| 7 || 0.5714 || 28 [https://catagolue.hatsya.com/object/xs28_db8n9arz3123032/b3s23 tub of snakes] || {{EmbedViewer|pname=7x7maxdensity |viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 1
|-
| 8 || 0.5625 || 36 nine blocks || {{EmbedViewer|pname=9blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 1
|-
| 9 || 0.5309 || 43 || {{EmbedViewer|pname=43cell9x9pseudostilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}} || 10
|-
| 10 || 0.5400 || 54 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=10x10maxdensity }} || >100
|-
| 11 || 0.5289 || 64 sixteen blocks|| {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=11x11maxdensity }} || 1
|-
| 12 || 0.5278 || 76 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=12x12maxdensity }}
|-
| 13 || 0.5325 || 90 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=13x13maxdensity }}
|-
| 14 || 0.5306 || 104 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=14x14maxdensity }}
|-
| 15 || 0.5289 || 119 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=15x15maxdensity }}
|-
| 16 || 0.531 || 136 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=16x16maxdensity }}
|-
| 17 || 0.526 || 152 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=17x17maxdensity }}
|-
| 18 || 0.528 || 171 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=18x18maxdensity }}
|-
| 19 || 0.526 || 190 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=19x19maxdensity }}
|-
| 20 || 0.525 || 210 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=20x20maxdensity }}
|-
| 21 || 0.526 || 232 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=21x21maxdensity }}
|-
| 22 || 0.523 || 253 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=22x22maxdensity }}
|-
| 23 || 0.522 || 276 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=23x23maxdensity }}
|-
| 24 || 0.523 || 301 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=24x24maxdensity }}
|-
| 25 || 0.522 || 326 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | pname=25x25maxdensity }}
|}

&dagger;After the first few terms, the numbers in this sequence closely approximate the triangular number sequence.<ref>{{LinkWikipedia|Triangular_number|name=Tringular number|nobullet=nobullet}}</ref>

*For any side length of the form 3n-1, a segment of [[block agar]] can be used to reach a population of 4n²; to n=4 this is the optimal solution, as can be seen in the above table.
*The maximum density of an nxn still life is greater than or equal to 1/2 for all finite n>1.<ref name="cgolmc_1">[[Conway's Game of Life: Mathematics and Construction]], Section 2.5-Still Life Density</ref>

== Densities of other objects ==
* The maximum density of a still life agar was proven to be 1/2 by [[Noam Elkies]]<ref>{{cite web|url=https://arxiv.org/abs/math/9905194 |title=The still-Life density problem and its generalizations|author=Noam D. Elkies| accessdate=May 26, 2022| date=May 31, 1999}}</ref> (this is known as the '''still life conjecture''').
* The maximum average density of an oscillating agar is conjectured but not proven to be 1/2. The current upper bound is 8/13, proven by [[Hartmut Holzwart]] in {{year|1992}}.<ref name="post147457" />
* The density of [[ash]] is [[Achim Flammenkamp's census|approximately 0.0287]] (1 in 35).
* By observation, the density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3.<ref name="post146282" />

==References==
<references>
<ref name="post147457">{{LinkForumThread
|format = ref
|title = Re: Can we substantiate this claim?
|p = 147457
|author = Hartmut Holzwart
|date = June 22, 2022
}}</ref>
<ref name="post146282">{{LinkForumThread
|format = ref
|title = Re: Can we substantiate this claim?
|p = 146282
|author = David Raucci
|date = May 29, 2022
}}</ref>
</references>

==External links==
{{LinkLexicon|lex_d.htm#density}}
*[https://golhobby.com/patman Maximum Density Still Lifes-PatMan pattern collection] all n=2...200
*[https://oeis.org/A055397 OEIS: A055397-Maximum population of an n X n stable pattern in Conway's Game of Life]

8life

2022-01-08T02:20:30Z

JsfasdF248: Changed redirect target from EightLife to OCA:EightLife

#REDIRECT [[OCA:EightLife]]

8life

2022-01-08T02:20:11Z

JsfasdF248: Redirected page to EightLife

#REDIRECT [[EightLife]]

Static symmetry

2021-10-02T16:55:07Z

JsfasdF248: /* Compositional/translational symmetry */

{{Ambox|text='''This article needs additional [[Help:Images|images]] for [[hexagonal neighbourhood|hexagonal]] and [[triangular neighbourhood|triangular]] symmetries.'''}}
The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd. Example: [[Long snake]]
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: [[Aircraft carrier]]
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even. Example: [[Snake]]

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|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"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|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 three possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===[[Gutter]] symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

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

===Compositional/translational symmetry===
For certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating a [[Margolus]]-neighbourhood rule.<ref name="post54857" /> One famous example of this is [[OCA:2×2|2×2]].

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

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

Most well-known [[wick]]s and [[agar]]s have translational symmetry, in one and two dimensions respectively, though some may also have reflectional symmetry, and agars can have rotational symmetry.

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.<ref>{{CiteHickersonOscillators|accessdate=May 2, 2019}}</ref>

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), 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_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
<ref name="post54857">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 54857
|author = Aidan F. Pierce
|date = January 8, 2018
}}</ref>
</references>

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

[[Category:Everything else]]

Static symmetry

2021-10-02T16:54:03Z

JsfasdF248: /* Compositional/translational symmetry */

{{Ambox|text='''This article needs additional [[Help:Images|images]] for [[hexagonal neighbourhood|hexagonal]] and [[triangular neighbourhood|triangular]] symmetries.'''}}
The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd. Example: [[Long snake]]
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: [[Aircraft carrier]]
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even. Example: [[Snake]]

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|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"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|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 three possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===[[Gutter]] symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

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

===Compositional/translational symmetry===
For certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating a [[Margolus]]-neighbourhood rule.<ref name="post54857" /> One famous example of this is [[OCA:2×2|2×2]].

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

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

Most well-known [[wick]]s and [[agar]]s have translational symmetry, in one and two dimensions respectively, though some may also have reflectional or rotational symmetry.

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.<ref>{{CiteHickersonOscillators|accessdate=May 2, 2019}}</ref>

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), 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_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
<ref name="post54857">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 54857
|author = Aidan F. Pierce
|date = January 8, 2018
}}</ref>
</references>

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

[[Category:Everything else]]

Static symmetry

2021-02-06T04:54:48Z

JsfasdF248: /* C4 */ Examples

{{Ambox|text='''This article needs additional [[Help:Images|images]] for [[hexagonal neighbourhood|hexagonal]] and [[triangular neighbourhood|triangular]] symmetries.'''}}
The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd. Example: [[Long snake]]
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: [[Aircraft carrier]]
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even. Example: [[Snake]]

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|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"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|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 three possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===[[Gutter]] symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

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

===Compositional/translational symmetry===
For certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating a [[Margolus]]-neighbourhood rule.<ref name="post54857" /> One famous example of this is [[OCA:2×2|2×2]].

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

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

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.<ref>{{CiteHickersonOscillators|accessdate=May 2, 2019}}</ref>

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), 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_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
<ref name="post54857">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 54857
|author = Aidan F. Pierce
|date = January 8, 2018
}}</ref>
</references>

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

[[Category:Everything else]]

Static symmetry

2021-02-06T04:36:49Z

JsfasdF248: /* C2 */ Examples

{{Ambox|text='''This article needs additional [[Help:Images|images]] for [[hexagonal neighbourhood|hexagonal]] and [[triangular neighbourhood|triangular]] symmetries.'''}}
The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd. Example: [[Long snake]]
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd. Example: [[Aircraft carrier]]
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even. Example: [[Snake]]

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|C2_4 symmetry]]
|}

====C4====
'''C4''': Symmetric under 90° rotation. There are two possibilities:

* '''C4_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''C4_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|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 three possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===[[Gutter]] symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

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

===Compositional/translational symmetry===
For certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating a [[Margolus]]-neighbourhood rule.<ref name="post54857" /> One famous example of this is [[OCA:2×2|2×2]].

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

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

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.<ref>{{CiteHickersonOscillators|accessdate=May 2, 2019}}</ref>

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), 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_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
<ref name="post54857">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 54857
|author = Aidan F. Pierce
|date = January 8, 2018
}}</ref>
</references>

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

[[Category:Everything else]]

Static symmetry

2021-02-06T04:30:17Z

JsfasdF248: /* C1 */

{{Ambox|text='''This article needs additional [[Help:Images|images]] for [[hexagonal neighbourhood|hexagonal]] and [[triangular neighbourhood|triangular]] symmetries.'''}}
The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd.
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|C2_4 symmetry]]
|}

====C4====
'''C4''': Symmetric under 90° rotation. There are two possibilities:

* '''C4_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''C4_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|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 three possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===[[Gutter]] symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

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

===Compositional/translational symmetry===
For certain rules, any pattern completely made of correctly-aligned arrangements of cells that fit within a 2×2 area will always be made up of said 2×2 arrangements of cells in all subsequent generations, simulating a [[Margolus]]-neighbourhood rule.<ref name="post54857" /> One famous example of this is [[OCA:2×2|2×2]].

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

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

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.<ref>{{CiteHickersonOscillators|accessdate=May 2, 2019}}</ref>

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), 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_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</ref>
<ref name="post66638">{{LinkForumThread
|format = ref
|title = Re: apgsearch v4.0
|p = 66638
|author = Adam P. Goucher
|date = December 20, 2018
}}</ref>
<ref name="post54857">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 54857
|author = Aidan F. Pierce
|date = January 8, 2018
}}</ref>
</references>

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

[[Category:Everything else]]

Static symmetry

2019-05-02T20:17:30Z

JsfasdF248: /* Oscillator symmetries */ Added sections

The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd.
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|C2_4 symmetry]]
|}

====C4====
'''C4''': Symmetric under 90° rotation. There are two possibilities:

* '''C4_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''C4_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|D4_x4 symmetry]]
|}

====D8====
'''D8''': Symmetric under both reflection and 90° rotation. The reflection symmetry will be with respect to horizontal, vertical, and diagonal lines. There are two possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===Gutter symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4). The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.


====Flippers====

====180 degree rotators====

====90 degree rotators====

===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), but these are not the same symmetries as square grids. C2, D2, and D4 symmetries are still compatible, but C4 symmetries become meaningless because the cells no longer have a side count that is divisible by 4. Other symmetries are exclusive to these alternative grids, as indicated below: 

* C1
* C2_1
* C2_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</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]]

Static symmetry

2019-05-02T02:38:33Z

JsfasdF248: Added sections for oscillators and spaceships (please expand)

The Life transition rule, like that of any [[Isotropic_non-totalistic_Life-like_cellular_automaton|isotropic cellular automaton]], is invariant under reflections and rotations. That is, the change in state of a cell remains the same if its [[neighbourhood]] is rotated or reflected. This implies there are '''symmetries''' which if present in a pattern are present in all its successors. Note that the converse is not true: a pattern need not have the full symmetry of one of its successor states.

==Square-grid symmetries==

[[File:Symmetries.png|center|frame|Overview of symmetries (excluding D8_2).]]

===Rotational symmetries===
Rotational symmetries include the following (note that "C" refers to the [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups]):

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C1.png|frame|C1 symmetry]]
|}

====C2====
'''C2''': Symmetric under 180° rotation. There are three possibilities:

* '''C2_1''': Rotation around the center of a cell. The bounding rectangle of a C2_1 pattern is odd by odd.
* '''C2_2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
* '''C2_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Symmetry C2_2.png|frame|C2_2 symmetry]]
| [[File:Symmetry C2_4.png|frame|C2_4 symmetry]]
|}

====C4====
'''C4''': Symmetric under 90° rotation. There are two possibilities:

* '''C4_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''C4_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry C4_1.png|frame|C4_1 symmetry]]
| [[File:Symmetry C4_4.png|frame|C4_4 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following (note that "D" refers to the [https://en.wikipedia.org/wiki/Dihedral_group dihedral groups]):

====D2====
'''D2''': Symmetric under reflection through a line. There are two possibilities:

* '''D2_+''' The line is orthogonal. There are two sub-possibilities:
** '''D2_+1''' The line bisects a row of cells. The bounding rectangle is odd by any.
** '''D2_+2''' The line lies between two rows of cells. The bounding rectangle is even by any.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1.png|frame|D2_+1 symmetry]]
| [[File:Symmetry D2_+2.png|frame|D2_+2 symmetry]]
|}

* '''D2_x''' The line is diagonal.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_x.png|frame|D2_x symmetry]]
|}

====D4====
'''D4''': Symmetric under both reflection and 180° rotation. The reflection symmetry will be with respect to two lines. There are two possibilities:

* '''D4_+''': The lines are orthogonal. There are three sub-possibilities:
** '''D4_+1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_+2''': Rotation around the midpoint of a side of a cell. The bounding rectangle is even by odd.
** '''D4_+4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1.png|frame|D4_+1 symmetry]]
| [[File:Symmetry D4_+2.png|frame|D4_+2 symmetry]]
| [[File:Symmetry D4_+4.png|frame|D4_+4 symmetry]]
|}

* '''D4_x''' The lines are diagonal. There are two sub-possibilities:
** '''D4_x1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
** '''D4_x4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Symmetry D4_x4.png|frame|D4_x4 symmetry]]
|}

====D8====
'''D8''': Symmetric under both reflection and 90° rotation. The reflection symmetry will be with respect to horizontal, vertical, and diagonal lines. There are two possibilities:

* '''D8_1''': Rotation around the center of a cell. The bounding rectangle is odd by odd.
* '''D8_2''': Rotation around a edge of a cell. The bounding rectangle is even by odd. This symmetry is not preserved by Life (reverting to D4_+2), but is with most bilaterally symmetric rules.
* '''D8_4''': Rotation around a corner of a cell. The bounding rectangle is even by even.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1.png|frame|D8_1 symmetry]]
| [[File:Symmetry D8_2.png|frame|D8_2 symmetry]]
| [[File:Symmetry D8_4.png|frame|D8_4 symmetry]]
|}

To preserve D8_2 symmetry, the following transitions must either all exist simultaneously with all other transitions in the same line or none should:<ref name="post28936" />

* B1c/B2c/B4c
* B1e/B2a/B2i/B4i
* B3e/B3j
* B3i/B6i
* B3q/B3y
* B4t/B5r
* B4e/B5y/B4w
* B4a/B5i
* B4n/B5e
* B5a/B7e/B8
* B6c/B6k
* S0/S1e/S3a
* S2c/S2k
* S2i/S5i
* S3e/S4r
* S3i/S4a/S4t/S6a/S6i/S7e
* S3r/S4i
* S3y/S4c/S4q
* S5e/S5j
* S5q/S5y
* S6c/S7c

===Skew symmetries===

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

===Gutter symmetries===

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

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

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_+1g.png|thumb|Orthogonal gutter symmetry D2_+1_gO1S0]]
| [[File:Symmetry D2_+1g s1.png|thumb|Orthogonal skewgutter symmetry D2_+1_gO1S1]]
| [[File:Symmetry D2_+1g s2.png|thumb|Orthogonal double skewgutter symmetry<ref name="post66043" /> D2_+1_gO1S2]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+1g.png|thumb|D4_+1 with orthogonal gutter symmetry D4_+1_gO1S0]]
| [[File:Symmetry D4_+1gg.png|thumb|D4_+1 with two orthogonal gutters D4_+1_gO1SO_gO1S0]]
| [[File:Symmetry D4_+2g.png|thumb|D4_+2 with orthogonal gutter symmetry D4_+2_gO1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2g s1.png|thumb|D4_+2 with orthogonal gutter symmetry, skewed]]
| [[File:Symmetry D4_+2g s2.png|thumb|D4_+2 with orthogonal gutter symmetry, doubly skewed]]
| [[File:Symmetry D4_+1twoskew.png|thumb|D4_+1 with two skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_+2 twoguttersoneskew.png|thumb|D4_+2 with two orthogonal gutters, one skewed]]
| [[File:Symmetry D4_+2 twoguttersonedoubleskew.png|thumb|D4_+2 with two orthogonal gutters, one doubly skewed]]
| [[File:Symmetry D4_+1twodoubleskew.png|thumb|D4_+1 with two double-skew orthogonal gutters]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D2_xg.png|thumb|Diagonal gutter symmetry D2_x_gD1S0]]
| [[File:Symmetry D2_xg s1.png|thumb|Diagonal skewgutter symmetry D2_x_gD1S1]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D4_x1g.png|thumb|D4_x1 with diagonal gutter symmetry D4_x1_gD1S0]]
| [[File:Symmetry D4_x1gg.png|thumb|D4_x1 with two diagonal gutters D4_x1_gD1S0_gD1S0]]
| [[File:Symmetry D4_x4g.png|thumb|D4_x4 with diagonal gutter symmetry D4_x4_gD1S0]]
| [[File:Symmetry D4_x4gg.png|thumb|D4_x4 with two diagonal gutters D4_x4_gD1S0_gD1S0]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1og.png|thumb|D8_1 with orthogonal gutter symmetry]]
| [[File:Symmetry D8_1dg.png|thumb|D8_1 with diagonal gutter symmetry]]
| [[File:Symmetry D8_1gg.png|thumb|D8_1 with orthogonal and diagonal gutters]]
| [[File:Symmetry D8_4g.png|thumb|D8_4 with diagonal gutter symmetry]]
|}

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Symmetry D8_1s1og.png|thumb|D8_1 with rotationally-symmetric orthogonal skewgutter symmetry]]
| [[File:Symmetry D8_1s1ogdg.png|thumb|and diagonal gutters]]
| [[File:Symmetry D8_1s2og.png|thumb|D8_1 with rotationally-symmetric orthogonal double skewgutter symmetry]]
| [[File:Symmetry D8_1s2ogdg.png|thumb|and diagonal gutters]]
|}

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

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

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

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

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

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

===Oscillator symmetries===

[[Oscillators]] have a wider range of symmetries than still lifes. This is because an oscillator can appear in two or four congruent states, not necessarily the same. There are 43 possible symmetries of oscillators, 27 of which do not appear in still lifes (i.e. period/mod = 2 or 4).



===Spaceship symmetries===

Spaceships have a limited range of symmetries because no spaceship can have rotational symmetry. However, spaceships can have glide symmetry, which is not applicable for finite patterns that do not move.

==Hexagonal-grid and triangular-grid symmetries==

[[hexagonal neighbourhood|Hexagonal]] and [[triangular neighbourhood|triangular]] grids have the same set of admissible symmetries as each other (by [http://mathworld.wolfram.com/DualTessellation.html planar duality]), but these are not the same symmetries as square grids. C2, D2, and D4 symmetries are still compatible, but C4 symmetries become meaningless because the cells no longer have a side count that is divisible by 4. Other symmetries are exclusive to these alternative grids, as indicated below: 

* C1
* C2_1
* C2_4
* 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" />

Hexagonal rules can also support gutter symmetry,<ref name="post66215" /> however, like with square grid gutters, apgsearch does not currently support searching with these.

===Rotational symmetries===
Rotational symmetries include the following:

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C1.png|frame|C1 symmetry]]
|}

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

* '''C2_1''': Rotation around the center of a cell.
* '''C2_4''': Rotation around a corner of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C2_1.png|frame|C2_1 symmetry]]
| [[File:Hexagonal symmetry C2_4.png|frame|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"
|-
| [[File:Hexagonal symmetry C3_1.png|frame|C3_1 symmetry]]
| [[File:Hexagonal symmetry C3_3.png|frame|C3_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry C6.png|frame|C6 symmetry]]
|}

===Reflectional symmetries===
Reflectional symmetries include the following:

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

* '''D2_x''': Through the vertices of a cell.
* '''D2_xo''': Through the edges of a cell.

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D2_x.png|frame|D2_x symmetry]]
| [[File:Hexagonal symmetry D2_xo.png|frame|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"
|-
| [[File:Hexagonal symmetry D4_x1.png|frame|D4_x1 symmetry]]
| [[File:Hexagonal symmetry D4_x4.png|frame|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 edges of cells.
* '''D6_1o''': Rotation around the center of a cell with lines going through the centers of cells.
* '''D6_3''': Rotation around the corner of a cell. (unsupported by apgsearch)

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D6_1.png|frame|D6_1 symmetry]]
| [[File:Hexagonal symmetry D6_1o.png|frame|D6_1o symmetry]]
| [[File:Hexagonal symmetry D6_3.png|frame|D6_3 symmetry]]
|}

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

{| style="margin-left: auto; margin-right: auto"
|-
| [[File:Hexagonal symmetry D12.png|frame|D12 symmetry]]
|}

==References==
<references>
<ref name="post28936">{{LinkForumThread
|format = ref
|title = Truly Bilateral Life-Like rules
|p = 28936
|author = BlinkerSpawn
|date = March 19, 2016
}}</ref>
<ref name="post66043">{{LinkForumThread
|format = ref
|title = Re: Rules with interesting dynamics
|p = 66043
|author = muzik
|date = November 30, 2018
}}</ref>
<ref name="post66215">{{LinkForumThread
|format = ref
|title = Re: Non-totalistic hex rules
|p = 66215
|author = muzik
|date = December 5, 2018
}}</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]]

Mod

2019-05-02T02:03:34Z

JsfasdF248: Added another type of p/m=2 symmetry

{{Glossary}}
The '''mod''' of an [[oscillator]] or [[spaceship]] is the smallest number of [[generation]]s that it takes for it to reappear in its original form, possibily subject to some rotation or reflection. The mod of a pattern may be equal to its [[period]], but it may also be a quarter of the period (for oscillators that [[:Category:Non-flipping oscillators that turn 90 degrees|rotate 90 degrees]] every quarter period) or half the period (for other oscillators that [[:Category:Non-flipping oscillators that turn 180 degrees|rotate 180 degrees]] every half period, oscillators with 180 degree rotational symmetry that rotate 90 degrees every half period, and also for [[flipper]]s).

==External links==
{{LinkLexicon|lex_m.htm#mod}}