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{{Glossary}}
{{Glossary}}
A '''reflectorless rotating oscillator''' (or '''looping spaceship''') - abbreviated as '''RRO''' - is a pattern that rotates itself after a certain number of [[generation]]s. There is the additional constraint that two non-interacting copies of the pattern could be combined into an [[oscillator]] with a [[period]] equal to exactly half of that of the component oscillators. This is like the [[pi orbital]], but without the stabilisation.
A '''reflectorless rotating oscillator''' (or '''looping spaceship''') - abbreviated as '''RRO''' - is a pattern that rotates itself after a certain number of [[generation]]s. An  additional constraint is that that it must be possible to arrange two or more non-interacting copies of the oscillator to produce a new [[oscillator]] with a lower [[period]]. This is analogous to the one- and two-pi variants of the [[pi orbital]], but without the stabilisation.


Such patterns have long been proven to exist (see [[universal constructor]]), but none were explicitly constructed in [[Life]] until the arrival of Adam P. Goucher's [[0E0P metacell]]. One of the isotropic-rule RROs listed below can be simulated using a matching arrangement of 0E0P metacells, and the result will closely resemble the chosen RRO. However, 0E0P metacells have an unalterable orientation, so the pattern after N generations never exactly matches a rotated copy of the original until the pattern returns to its original configuration. To rectify this, four disjoint copies of the resulting oscillator, each one quarter-phase apart, can be arranged into a configuration which undergoes a 90-degree rotation every quarter of its period. This still retains the loopability constraint necessary for a classical RRO.
A reflectorless rotating oscillator has a '''loopability''' of ''n'', if ''n'' independent oscillators can follow the exact same path, in a way that reduces the period of the combined oscillator by a factor of ''n''. Along similar lines, the '''multiplicity''' of an RRO is its ''maximum'' loopability number. Another way of stating the additional constraint in the previous paragraph is that a "classical" RRO must have loopability greater than 1.


Alternatively, the term may refer to any statorless oscillator that rotates itself after a certain number of generations. The term "statorless rotating oscillator" is sometimes used to refer to these, as opposed to "classical" RROs. [[Blinker]]s or [[monogram]]s could technically be considered to be degenerate rotating oscillators, turning 90 degrees on every half-period.  Rotationally symmetric objects such as [[clock II]] (or any oscillator where period = 4*mod, such as [[pinwheel]] and [[sixty-nine]]) are certainly rotating oscillators.  However, most such patterns have stator cells, and more importantly they do not fulfill the additional constraint of allowing for two or more non-interacting copies of the rotor, so they are definitely not "loopable" RROs.
== First known examples in Life ==
Reflectorless rotating oscillator patterns have long been known to exist (see [[universal constructor]]), but none were explicitly constructed in [[Life]] until the arrival of Adam P. Goucher's [[0E0P metacell]]. Any of the isotropic-rule RROs listed below can be simulated using a matching arrangement of 0E0P metacells, and the result will closely resemble the chosen RRO.


Special-purpose true RROs could be constructed using known universal-constructor technology, which would be much simpler and lower population than a single-loop pseudo-RRO or multi-loop true RRO based on 0E0P metacells. A universal constructor-based RRO has no limit on the number of independent patterns that can orbit a single point.
However, 0E0P metacells have an unalterable orientation, so the pattern after N generations never exactly matches a rotated copy of the original until the pattern returns to its original configuration. To rectify this, four disjoint copies of the resulting oscillator, each one quarter-phase apart, can be arranged into a configuration which undergoes a 90-degree rotation every quarter of its period. This still retains the loopability constraint necessary for a classical RRO.


==Other rules==
=== Engineered RROs ===
Patterns much simpler than 0E0P metacells can also demonstrate RRO behavior. In August 2021, [[Goldtiger997]] completed a reflectorless rotating oscillator consisting mostly of a long string of [[single-channel]] gliders.<ref name="post135135" /> These gliders represent two copies of a construction recipe for the four corners of a diamond-shaped signal loop. The first string of recipe gliders builds the loop, which catches the second string of gliders and then sends out two copies of them in a new direction, before self-destructing.


===Outer-totalistic rules===
Universal constructor-based RROs have no theoretical limit on the number of independent oscillators that can orbit a single point, but any specific RRO will have a maximum loopability value. For example, Goldtiger997's engineered RRO has a multiplicity of 11.<ref name="post135146" />


There is only one known [[elementary]] reflectorless rotating oscillator in an outer-totalistic [[Life-like cellular automaton]]. It exists in B02348/S0123, and has a period of 272. It is technically not a classical RRO, because two copies combined into a half-period oscillator interact but do not interfere with each other.
{{POTY|name=Goldtiger997's RRO in Conway's Game of Life|year=2021|rank=2|behind=the [[self-synthesizing oblique loopship]]}} See the [[SSOL]] article for more details about a self-constructing memory loop that uses the same technology.


<gallery>
== The loopability-1 case ==
File:Xp272 757w9.png|Reflectorless rotating oscillator in B02348/S0123.<br />{{LinkCatagolue|xp272_757w9|rule=b02348s0123|style=brief}}<br />Twofold: {{LinkCatagolue|xp136_w1ee5jzzgzca778|rule=b02348s0123|style=brief}}
The term '''RRO''' is occasionally used to refer to any statorless oscillator that rotates itself after a certain number of generations. The term "statorless rotating oscillator" is occasionally used as an alternate term for these "loopability-1" RROs,<ref name="post32201" /><ref name="post86494" /> to distinguish them from classical loopability>1 RROs.
</gallery>
 
Rotationally symmetric objects such as [[clock II]] (or any oscillator where period = 4*mod, such as [[pinwheel]] and [[sixty-nine]]) are certainly rotating oscillators, and they contain no reflectors, so they fit the description "reflectorless rotating oscillator" but not the above technical definition of an RRO. Most such patterns have stator cells, and they certainly don't allow for two or more non-interacting copies of the rotor, so they are definitely not "loopable" statorless RROs.
 
A related category of loopability-1 oscillator is the '''reflectorless flipping oscillator''', or '''RFO''',<ref name="post47132" /> where a statorless oscillator evolves into its own mirror image. Such oscillators can't allow a loopability of 2 for obvious reasons; two such oscillators following the same path would have to collide onto each other.  RFOs with odd loopability greater than 1 are theoretically possible,<ref name="post193128" /> but there are currently no known examples in Conway's Life.
 
Patterns such as the p160 oscillator in {{rl|tlife}}, the p32 ("spinner") in {{rl|DryLife}}, and the p88 in B36ce7c/S23-y could be called loopability-1 RROs, and the natural p424 gun in {{rl|Pedestrian Life}} is a loopability-1 ''RRG'' (reflectorless rotating gun).
 
== Other rules ==
 
=== Outer-totalistic rules ===
[[File:Xp272 757w9.png|thumb|right|Reflectorless rotating oscillator in B02348/S0123.<br />{{LinkCatagolue|xp272_757w9|rule=b02348s0123|style=brief}}<br />Twofold: {{LinkCatagolue|xp136_w1ee5jzzgzca778|rule=b02348s0123|style=brief}}]]
 
Aside from instances like the Pedestrian Life p424 gun, there is only one known [[elementary]] reflectorless rotating oscillator in a [[Life-like cellular automaton]]. It exists in B02348/S0123, and has a period of 272. It is technically not a classical RRO, because two copies combined into a half-period oscillator interact but do not interfere with each other.
 
=== Isotropic non-totalistic rules ===
[[File:Xp420 59qkxs.png|thumb|left|Reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7.<br />{{LinkCatagolue|xp420_59qkxs|rule=b2i34ik7s23-a4ikn5j7|style=brief}}]]


===Non-totalistic rules===
Multiple reflectorless rotating oscillators have been found in [[isotropic non-totalistic]] rules, especially recently:
Multiple reflecterless rotating oscillators have been found in [[non-totalistic]] rules, especially recently:
* An early example in the exploration of isotropic non-totalistic rules is a period-168 case in the rule plife2, noted in November 2015.<ref>https://conwaylife.com/forums/viewtopic.php?p=24901#p24901</ref>
* dmqwerty425 discovered a period-420 reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7 on Catagolue in November 2016.<ref name="post37225" />
* dmqwerty425 discovered a period-420 reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7 on Catagolue in November 2016.<ref name="post37225" />
* Daniel R. Collazo discovered a period-184 reflectorless rotating oscillator in B3/S23-a4i5i6ci in July 2017.<ref name="post46748" />
* dani discovered a period-184 reflectorless rotating oscillator in B3/S23-a4i5i6ci in July 2017.<ref name="post46748" />
* Rhombic discovered a period-72 reflectorless rotating oscillator in B2e3-a4a/S1c23-aky in August 2017.<ref name="post48366" />
* Rhombic discovered a period-72 reflectorless rotating oscillator in B2e3-a4a/S1c23-aky in August 2017.<ref name="post48366" />
* Saka discovered a period-68 reflectorless rotating oscillator in B3-n4rtw5i/S23-n4q5i in August 2017.<ref name="post48513" />
* Saka discovered a period-68 reflectorless rotating oscillator in B3-n4rtw5i/S23-n4q5i in August 2017.<ref name="post48513" />
* 2718281828 discovered a number of loopable RROs, with 2-, 3-, 4-, 5-, and even 6-fold loopability, in July 2018.<ref name="post62035" />
* [[2718281828]] discovered a number of loopable RROs, with 2-, 3-, 4-, 5-, and even 6-fold loopability, in July 2018.<ref name="post62035" />
* [[AforAmpere]] discovered a 10-fold RRO, period 500, in B3aeijr4jz5ckr6cin7c8/S2-i3-ak4ceinrtz5cej6ce7c8. 15 copies of the oscillator will fit in the same loop, but without reducing the period.<ref name="post78513" />
* AforAmpere also discovered an RRO with 11-fold loopability in B3-cnqy4j5ckr6cn8/S2-i3-ak4ceinrtz5cj6ci7c8, allowing the period 616 oscillator to be reduced to period 56.<ref name="post78513" />
 
Another wave of record-breaking highly loopable reflectorless rotating oscillators followed in mid-July 2019, namely:
 
* 2718281828 discovered a {1,2,4,5,10,20}-loopable RRO in B2n3-k4cj5ek67/S2n3-jk4acikrty5cr678 on 16th July, followed by a {1,2,4,7,14,28}-loopable RRO in B2n3-k4jq5k67/S2n3-jy4aikrty5cr678 and a {1,2,3,4,6,9,12,18,36}-loopable RRO in B3-k4cejqz5e6-e78/S2n3-jk4-cjnw5c678.
* AforAmpere discovered a {2,3,4,6,12,17,19,34,38,51,57}-loopable RRO in B34cjq5e6-en78/S2n3-jk4-jnqw5ceky678.
* 2718281828 discovered a {1,2,3,4,6,12,13,26,29,39,52,58}-loopable RRO in B2in3-kq4cjy5eky6-e78/S3-jk4-cjnw5ce678.
* AforAmpere discovered a {1,2,4,17,34,68,89}-loopable RRO in B2i3-k4cjqyz5ey6-en78/S2n3-j4-jnwy5cky678.
 
The last of these, for example, makes it trivial to construct an 89-loopable RRO in B3/S23 using the [[0E0P metacell]].
 
A family of adjustable reflectorless rotating oscillators was found not too long ago.
 
=== Larger than Life ===
In June {{year|2001}} Dean Hickerson discovered a reflectorless rotating oscillator with a period of 552 in a [[Larger than Life]] rule. He placed eight copies in a circle, yielding a period-69 oscillator. In November {{year|2002}} [[Dave Greene]] noticed that twelve copies can orbit a central point with period 46.<ref name="post43299" />


<gallery>
<gallery>
File:Xp420 59qkxs.png|Reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7.<br />{{LinkCatagolue|xp420_59qkxs|rule=b2i34ik7s23-a4ikn5j7|style=brief}}
Deanhickersons p552 RRO.png|Dean Hickerson's period-552 reflectorless rotating oscillator in the Larger than Life rule R7,C0,M1,S65..114,B65..95,NM.<br>{{LinkCatagolue|xp552_0gsuvvvvvuszevvv33133vvvezw377777733|rule=r7b65t95s65t114|style=brief}}
Deanhickersons p552 RRO 12fold p46.png|Dave Greene's 12-fold, period-46 version of Dean Hickerson's period-552 RRO.
</gallery>
</gallery>


===Larger than Life===
=== Hexagonal grid ===
One reflectorless rotating oscillator has been found in a [[Larger than Life]] rule, discovered by Dean Hickerson with a period of 552. He placed eight copies in a circle, yielding a period-69 oscillator. [[Dave Greene]] noticed that twelve copies can orbit a central point with period 46.<ref name="post43299" />
There are currently five known reflectorless rotating oscillators that work in isotropic non-totalistic hexagonal rules so far. All five of these rotate 60 degrees around a central hexagon. There are currently no known loopable oscillators which rotate 120 degrees around a central cell or vertex, though examples must exist.
{| class="wikitable" style="text-align:center"
| {{EmbedViewer
|rle          = x = 3, y = 2, rule = B2o3o4m5/S2H
3o$2bo!
|viewerconfig = #C [[ STARTFROM 30 THEME Book AUTOSTART GPS 5 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}
| {{EmbedViewer
|rle          = x = 8, y = 8, rule = B2-m4-o/S23p4m5H
2o$3bo$3bo$3b2o3$6bo$6b2o!
|viewerconfig = #C [[ STARTFROM 228 THEME Book AUTOSTART GPS 38 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}
| {{EmbedViewer
|rle          = x = 3, y = 5, rule = B2-p3o4-o6/S2m3-oH
bo$bo$b2o$3o$2bo!
|viewerconfig = #C [[ STARTFROM 324 THEME Book AUTOSTART GPS 54 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}
| {{EmbedViewer
|rle          = x = 4, y = 2, rule = B2o3-o4m6/S2-o34p5H
4o$o2bo!
|viewerconfig = #C [[ STARTFROM 342 THEME Book AUTOSTART GPS 57 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}
| {{EmbedViewer
|rle          = x = 5, y = 5, rule = B2-m4-m56/S2m34m56H
3bo$ob3o$3o$bobo$b3o!
|viewerconfig = #C [[ STARTFROM 594 THEME Book AUTOSTART GPS 66 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}
|-
| p30, up to 3<ref>https://conwaylife.com/forums/viewtopic.php?p=73919#p73919</ref>
| p228, up to 4<ref name="manyhexrros">https://conwaylife.com/forums/viewtopic.php?p=220497#p220497</ref>
| p324, up to 2<ref name="manyhexrros"/>
| p342, up to 3<ref name="manyhexrros"/>
| p594, up to 3<ref>found no later than 2012: https://conwaylife.com/forums/viewtopic.php?p=6823#p6823</ref>
|}


<gallery>
=== Triangular grid ===
File:Deanhickersons p552 RRO.png|Dean Hickerson's period-552 reflectorless rotating oscillator in the [[Larger than Life]] rule R7,C0,M1,S65..114,B65..95,NM.<br>{{LinkCatagolue|code=xp552_0gsuvvvvvuszevvv33133vvvezw377777733|rule=r7b65t95s65t114|style=brief}}
There are no known examples in range-1 rules, but one is known in a range-2 weighted rule, of which two can fit in a loop:
File:Deanhickersons p552 RRO 12fold p46.png|Dave Greene's 12-fold, period-46 version of Dean Hickerson's period-552 RRO.
{{EmbedViewer
</gallery>
|position    = center
|rle          = x = 21, y = 10, rule = R2,C2,S2,5,8,11,15,19,22-26,29,32,43,46-47,49-50,52-53,55-56,67,70-71,73-74,76-77,79-80,82-83,91,94-95,97-98,100-101,103-104,106-107,109-111,B3,6,9,12,27,30,33,36,51,54,57,60,75,78,81,84,99,102,105,108,NW000000000001041c0401041c001c0400040104000000000000L
$4bo5bob2o3b2obo$b2o2bo$o$9bo$o7b2o$10bo$6bo4bo2$9b3o!
|viewerconfig = #C [[ STARTFROM 444 THEME Book AUTOSTART GPS 74 AUTOFIT HISTORYFIT ]]
}}


==Classification==
No examples are known of loopable oscillators which rotate 120 degrees around a triangular cell center or a vertex.
A method for classifying RROs by the amount of times they can fit into a single loop in a way that evenly divides the period has been discussed;<ref>http://conwaylife.com/forums/viewtopic.php?f=11&t=2238&p=47086&hilit=rro#p47086</ref> by this logic, patterns such as the p160 oscillator in [[tlife]], the p32 in B378/S23, the p88 in B36ce7c/S23-y, and the natural gun in [[Pedestrian Life]] could be classed as reflectorless rotating oscillators with a ''loopability'' of 1.


==References==
== References ==
<references>
<references>
<ref name="post37225">{{LinkForumThread
<ref name="post37225">{{LinkForumThread
Line 47: Line 117:
|t      = 1638
|t      = 1638
|p      = 37225
|p      = 37225
|title  = Soup search results in rules other than Conway's Life
|title  = Re: Soup search results in rules other than Conway's Life
|format = ref
|format = ref
|author = praosylen
|date  = November 10, 2016
}}</ref>
}}</ref>
<ref name="post43299">{{LinkForumThread
<ref name="post43299">{{LinkForumThread
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|t      = 2853
|t      = 2853
|p      = 43299
|p      = 43299
|title  = Abstract Art
|title  = Re: Abstract Art
|format = ref
|format = ref
|author = Dave Greene
|date  = May 7, 2017
}}</ref>
}}</ref>
<ref name="post46748">{{LinkForumThread
<ref name="post46748">{{LinkForumThread
Line 61: Line 135:
|t      = 2036
|t      = 2036
|p      = 46748
|p      = 46748
|title  = Thread for basic questions
|title  = Re: Thread for basic questions
|format = ref
|format = ref
|author = dani
|date  = July 16, 2017
}}</ref>
}}</ref>
<ref name="post48366">{{LinkForumThread
<ref name="post48366">{{LinkForumThread
Line 68: Line 144:
|t      = 1971
|t      = 1971
|p      = 48366
|p      = 48366
|title  = Miscellaneous Discoveries in Other Cellular Automata
|title  = Re: Miscellaneous Discoveries in Other Cellular Automata
|format = ref
|format = ref
|author = Rhombic
|date  = August 8, 2017
}}</ref>
}}</ref>
<ref name="post48513">{{LinkForumThread
<ref name="post48513">{{LinkForumThread
Line 75: Line 153:
|t      = 376
|t      = 376
|p      = 48513
|p      = 48513
|title  = Thread for Your Accidental Discoveries that Aren't in CGOL
|title  = Re: Thread for Your Accidental Discoveries that Aren't in CGOL
|format = ref
|format = ref
|author = Saka
|date  = August 10, 2017
}}</ref>
}}</ref>
<ref name="post62035">{{LinkForumThread
<ref name="post62035">{{LinkForumThread
Line 82: Line 162:
|t      = 3447
|t      = 3447
|p      = 62035
|p      = 62035
|title  = Reflectorless Rotating Oscillators (RRO)
|title  = Re: Reflectorless Rotating Oscillators (RRO)
|format = ref
|author = 2718281828
|date  = July 20, 2018
}}</ref>
<ref name="post78513">{{LinkForumThread
|format = ref
|title  = Re: Reflectorless Rotating Oscillators (RRO)
|p      = 78513
|author = AforAmpere
|date  = June 27, 2019
}}</ref>
<ref name="post135135">{{LinkForumThread
|p      = 135135
|title  = Re: Reflectorless Rotating Oscillator Discussion Thread in Life
|format = ref
|author = Goldtiger997
|date  = August 21, 2021
}}</ref>
<ref name="post32201">{{LinkForumThread
|p      = 32201
|title  = Re: Thread for your basic questions
|format = ref
|author = muzik
|date  = June 23, 2016
}}</ref>
<ref name="post47132">{{LinkForumThread
|p      = 47132
|title  = Re: Thread for basic non-CGOL questions
|format = ref
|author = Connor Steppie
|date  = July 23rd, 2017
}}</ref>
<ref name="post86494">{{LinkForumThread
|p      = 86494
|title  = Re: MaritimeLife (B2i34c6cen7c8/S2-i3-q4c5a6e8)
|format = ref
|author = Entity Valkyrie
|date  = December 23, 2019
}}</ref>
<ref name="post135146">{{LinkForumThread
|p      = 135146
|title  = Re: Reflectorless Rotating Oscillator Discussion Thread in Life
|format = ref
|author = Adam P. Goucher
|date  = August 21st, 2021
}}</ref>
<ref name="post193128">{{LinkForumThread
|p      = 135146
|title  = Re: Can we substantiate this claim?
|format = ref
|format = ref
|author = confocaloid
|date  = August 28st, 2024
}}</ref>
}}</ref>
</references>
</references>
==External links==
{{LinkForumThread|f=2|t=393|title=Rotary oscillators?}}


== External links ==
* {{LinkLexicon|lex_r.htm#reflectorlessrotatingoscillator}}
* {{LinkLexicon|filename=lex_m.htm#multiplicity|patternname=Multiplicity}}
* {{LinkForumThread|f=2|t=393|title=Rotary oscillators?}}
* {{LinkForumThread|f=11|t=3447|title=Reflectorless Rotating Oscillators (RRO)}}


__NOTOC__
__NOTOC__

Latest revision as of 10:17, 15 July 2026

A reflectorless rotating oscillator (or looping spaceship) - abbreviated as RRO - is a pattern that rotates itself after a certain number of generations. An additional constraint is that that it must be possible to arrange two or more non-interacting copies of the oscillator to produce a new oscillator with a lower period. This is analogous to the one- and two-pi variants of the pi orbital, but without the stabilisation.

A reflectorless rotating oscillator has a loopability of n, if n independent oscillators can follow the exact same path, in a way that reduces the period of the combined oscillator by a factor of n. Along similar lines, the multiplicity of an RRO is its maximum loopability number. Another way of stating the additional constraint in the previous paragraph is that a "classical" RRO must have loopability greater than 1.

First known examples in Life

Reflectorless rotating oscillator patterns have long been known to exist (see universal constructor), but none were explicitly constructed in Life until the arrival of Adam P. Goucher's 0E0P metacell. Any of the isotropic-rule RROs listed below can be simulated using a matching arrangement of 0E0P metacells, and the result will closely resemble the chosen RRO.

However, 0E0P metacells have an unalterable orientation, so the pattern after N generations never exactly matches a rotated copy of the original until the pattern returns to its original configuration. To rectify this, four disjoint copies of the resulting oscillator, each one quarter-phase apart, can be arranged into a configuration which undergoes a 90-degree rotation every quarter of its period. This still retains the loopability constraint necessary for a classical RRO.

Engineered RROs

Patterns much simpler than 0E0P metacells can also demonstrate RRO behavior. In August 2021, Goldtiger997 completed a reflectorless rotating oscillator consisting mostly of a long string of single-channel gliders.[1] These gliders represent two copies of a construction recipe for the four corners of a diamond-shaped signal loop. The first string of recipe gliders builds the loop, which catches the second string of gliders and then sends out two copies of them in a new direction, before self-destructing.

Universal constructor-based RROs have no theoretical limit on the number of independent oscillators that can orbit a single point, but any specific RRO will have a maximum loopability value. For example, Goldtiger997's engineered RRO has a multiplicity of 11.[2]

Goldtiger997's RRO in Conway's Game of Life ranked second place in the Pattern of the Year 2021 competition on the ConwayLife.com forums, behind the self-synthesizing oblique loopship.[3] See the SSOL article for more details about a self-constructing memory loop that uses the same technology.

The loopability-1 case

The term RRO is occasionally used to refer to any statorless oscillator that rotates itself after a certain number of generations. The term "statorless rotating oscillator" is occasionally used as an alternate term for these "loopability-1" RROs,[4][5] to distinguish them from classical loopability>1 RROs.

Rotationally symmetric objects such as clock II (or any oscillator where period = 4*mod, such as pinwheel and sixty-nine) are certainly rotating oscillators, and they contain no reflectors, so they fit the description "reflectorless rotating oscillator" but not the above technical definition of an RRO. Most such patterns have stator cells, and they certainly don't allow for two or more non-interacting copies of the rotor, so they are definitely not "loopable" statorless RROs.

A related category of loopability-1 oscillator is the reflectorless flipping oscillator, or RFO,[6] where a statorless oscillator evolves into its own mirror image. Such oscillators can't allow a loopability of 2 for obvious reasons; two such oscillators following the same path would have to collide onto each other. RFOs with odd loopability greater than 1 are theoretically possible,[7] but there are currently no known examples in Conway's Life.

Patterns such as the p160 oscillator in tlife, the p32 ("spinner") in DryLife, and the p88 in B36ce7c/S23-y could be called loopability-1 RROs, and the natural p424 gun in Pedestrian Life is a loopability-1 RRG (reflectorless rotating gun).

Other rules

Outer-totalistic rules

Reflectorless rotating oscillator in B02348/S0123.
Catagolue: here
Twofold: Catagolue: here

Aside from instances like the Pedestrian Life p424 gun, there is only one known elementary reflectorless rotating oscillator in a Life-like cellular automaton. It exists in B02348/S0123, and has a period of 272. It is technically not a classical RRO, because two copies combined into a half-period oscillator interact but do not interfere with each other.

Isotropic non-totalistic rules

Reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7.
Catagolue: here

Multiple reflectorless rotating oscillators have been found in isotropic non-totalistic rules, especially recently:

  • An early example in the exploration of isotropic non-totalistic rules is a period-168 case in the rule plife2, noted in November 2015.[8]
  • dmqwerty425 discovered a period-420 reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7 on Catagolue in November 2016.[9]
  • dani discovered a period-184 reflectorless rotating oscillator in B3/S23-a4i5i6ci in July 2017.[10]
  • Rhombic discovered a period-72 reflectorless rotating oscillator in B2e3-a4a/S1c23-aky in August 2017.[11]
  • Saka discovered a period-68 reflectorless rotating oscillator in B3-n4rtw5i/S23-n4q5i in August 2017.[12]
  • 2718281828 discovered a number of loopable RROs, with 2-, 3-, 4-, 5-, and even 6-fold loopability, in July 2018.[13]
  • AforAmpere discovered a 10-fold RRO, period 500, in B3aeijr4jz5ckr6cin7c8/S2-i3-ak4ceinrtz5cej6ce7c8. 15 copies of the oscillator will fit in the same loop, but without reducing the period.[14]
  • AforAmpere also discovered an RRO with 11-fold loopability in B3-cnqy4j5ckr6cn8/S2-i3-ak4ceinrtz5cj6ci7c8, allowing the period 616 oscillator to be reduced to period 56.[14]

Another wave of record-breaking highly loopable reflectorless rotating oscillators followed in mid-July 2019, namely:

  • 2718281828 discovered a {1,2,4,5,10,20}-loopable RRO in B2n3-k4cj5ek67/S2n3-jk4acikrty5cr678 on 16th July, followed by a {1,2,4,7,14,28}-loopable RRO in B2n3-k4jq5k67/S2n3-jy4aikrty5cr678 and a {1,2,3,4,6,9,12,18,36}-loopable RRO in B3-k4cejqz5e6-e78/S2n3-jk4-cjnw5c678.
  • AforAmpere discovered a {2,3,4,6,12,17,19,34,38,51,57}-loopable RRO in B34cjq5e6-en78/S2n3-jk4-jnqw5ceky678.
  • 2718281828 discovered a {1,2,3,4,6,12,13,26,29,39,52,58}-loopable RRO in B2in3-kq4cjy5eky6-e78/S3-jk4-cjnw5ce678.
  • AforAmpere discovered a {1,2,4,17,34,68,89}-loopable RRO in B2i3-k4cjqyz5ey6-en78/S2n3-j4-jnwy5cky678.

The last of these, for example, makes it trivial to construct an 89-loopable RRO in B3/S23 using the 0E0P metacell.

A family of adjustable reflectorless rotating oscillators was found not too long ago.

Larger than Life

In June 2001 Dean Hickerson discovered a reflectorless rotating oscillator with a period of 552 in a Larger than Life rule. He placed eight copies in a circle, yielding a period-69 oscillator. In November 2002 Dave Greene noticed that twelve copies can orbit a central point with period 46.[15]

  • Dean Hickerson's period-552 reflectorless rotating oscillator in the Larger than Life rule R7,C0,M1,S65..114,B65..95,NM. Catagolue: here
    Dean Hickerson's period-552 reflectorless rotating oscillator in the Larger than Life rule R7,C0,M1,S65..114,B65..95,NM.
    Catagolue: here
  • Dave Greene's 12-fold, period-46 version of Dean Hickerson's period-552 RRO.
    Dave Greene's 12-fold, period-46 version of Dean Hickerson's period-552 RRO.

Hexagonal grid

There are currently five known reflectorless rotating oscillators that work in isotropic non-totalistic hexagonal rules so far. All five of these rotate 60 degrees around a central hexagon. There are currently no known loopable oscillators which rotate 120 degrees around a central cell or vertex, though examples must exist.

x = 3, y = 2, rule = B2o3o4m5/S2H 3o$2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 30 THEME Book AUTOSTART GPS 5 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
x = 8, y = 8, rule = B2-m4-o/S23p4m5H 2o$3bo$3bo$3b2o3$6bo$6b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 228 THEME Book AUTOSTART GPS 38 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
x = 3, y = 5, rule = B2-p3o4-o6/S2m3-oH bo$bo$b2o$3o$2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 324 THEME Book AUTOSTART GPS 54 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
x = 4, y = 2, rule = B2o3-o4m6/S2-o34p5H 4o$o2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 342 THEME Book AUTOSTART GPS 57 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
x = 5, y = 5, rule = B2-m4-m56/S2m34m56H 3bo$ob3o$3o$bobo$b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 594 THEME Book AUTOSTART GPS 66 AUTOFIT HISTORYFIT THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
p30, up to 3[16] p228, up to 4[17] p324, up to 2[17] p342, up to 3[17] p594, up to 3[18]

Triangular grid

There are no known examples in range-1 rules, but one is known in a range-2 weighted rule, of which two can fit in a loop:

x = 21, y = 10, rule = R2,C2,S2,5,8,11,15,19,22-26,29,32,43,46-47,49-50,52-53,55-56,67,70-71,73-74,76-77,79-80,82-83,91,94-95,97-98,100-101,103-104,106-107,109-111,B3,6,9,12,27,30,33,36,51,54,57,60,75,78,81,84,99,102,105,108,NW000000000001041c0401041c001c0400040104000000000000L $4bo5bob2o3b2obo$b2o2bo$o$9bo$o7b2o$10bo$6bo4bo2$9b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ STARTFROM 444 THEME Book AUTOSTART GPS 74 AUTOFIT HISTORYFIT ]]
(click above to open LifeViewer)

No examples are known of loopable oscillators which rotate 120 degrees around a triangular cell center or a vertex.

References

  1. ↑ Goldtiger997 (August 21, 2021). Re: Reflectorless Rotating Oscillator Discussion Thread in Life (discussion thread) at the ConwayLife.com forums
  2. ↑ Adam P. Goucher (August 21st, 2021). Re: Reflectorless Rotating Oscillator Discussion Thread in Life (discussion thread) at the ConwayLife.com forums
  3. ↑ Pavel Grankovskiy (April 6, 2022). Re: PotY 2020 Voting Thread (discussion thread) at the ConwayLife.com forums
  4. ↑ muzik (June 23, 2016). Re: Thread for your basic questions (discussion thread) at the ConwayLife.com forums
  5. ↑ Entity Valkyrie (December 23, 2019). Re: MaritimeLife (B2i34c6cen7c8/S2-i3-q4c5a6e8) (discussion thread) at the ConwayLife.com forums
  6. ↑ Connor Steppie (July 23rd, 2017). Re: Thread for basic non-CGOL questions (discussion thread) at the ConwayLife.com forums
  7. ↑ confocaloid (August 28st, 2024). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
  8. ↑ https://conwaylife.com/forums/viewtopic.php?p=24901#p24901
  9. ↑ praosylen (November 10, 2016). Re: Soup search results in rules other than Conway's Life (discussion thread) at the ConwayLife.com forums
  10. ↑ dani (July 16, 2017). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
  11. ↑ Rhombic (August 8, 2017). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
  12. ↑ Saka (August 10, 2017). Re: Thread for Your Accidental Discoveries that Aren't in CGOL (discussion thread) at the ConwayLife.com forums
  13. ↑ 2718281828 (July 20, 2018). Re: Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums
  14. ↑ 14.0 14.1 AforAmpere (June 27, 2019). Re: Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums
  15. ↑ Dave Greene (May 7, 2017). Re: Abstract Art (discussion thread) at the ConwayLife.com forums
  16. ↑ https://conwaylife.com/forums/viewtopic.php?p=73919#p73919
  17. ↑ 17.0 17.1 17.2 https://conwaylife.com/forums/viewtopic.php?p=220497#p220497
  18. ↑ found no later than 2012: https://conwaylife.com/forums/viewtopic.php?p=6823#p6823


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