Reflectorless rotating oscillator
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
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
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.
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]
References
- ↑ Goldtiger997 (August 21, 2021). Re: Reflectorless Rotating Oscillator Discussion Thread in Life (discussion thread) at the ConwayLife.com forums
- ↑ Adam P. Goucher (August 21st, 2021). Re: Reflectorless Rotating Oscillator Discussion Thread in Life (discussion thread) at the ConwayLife.com forums
- ↑ Pavel Grankovskiy (April 6, 2022). Re: PotY 2020 Voting Thread (discussion thread) at the ConwayLife.com forums
- ↑ muzik (June 23, 2016). Re: Thread for your basic questions (discussion thread) at the ConwayLife.com forums
- ↑ Entity Valkyrie (December 23, 2019). Re: MaritimeLife (B2i34c6cen7c8/S2-i3-q4c5a6e8) (discussion thread) at the ConwayLife.com forums
- ↑ Connor Steppie (July 23rd, 2017). Re: Thread for basic non-CGOL questions (discussion thread) at the ConwayLife.com forums
- ↑ confocaloid (August 28st, 2024). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
- ↑ https://conwaylife.com/forums/viewtopic.php?p=24901#p24901
- ↑ praosylen (November 10, 2016). Re: Soup search results in rules other than Conway's Life (discussion thread) at the ConwayLife.com forums
- ↑ dani (July 16, 2017). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
- ↑ Rhombic (August 8, 2017). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
- ↑ Saka (August 10, 2017). Re: Thread for Your Accidental Discoveries that Aren't in CGOL (discussion thread) at the ConwayLife.com forums
- ↑ 2718281828 (July 20, 2018). Re: Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums
- ↑ 14.0 14.1 AforAmpere (June 27, 2019). Re: Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums
- ↑ Dave Greene (May 7, 2017). Re: Abstract Art (discussion thread) at the ConwayLife.com forums
External links
- Reflectorless rotating oscillator at the Life Lexicon
- Multiplicity at the Life Lexicon
- Rotary oscillators? (discussion thread) at the ConwayLife.com forums
- Reflectorless Rotating Oscillators (RRO) (discussion thread) at the ConwayLife.com forums