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. 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.

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.

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. These gliders represent two copies of a single-channel recipe for the construction of 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. See the SSOL article for more details about a self-constructing memory loop that uses the same technology.

Alternatively, the term RRO 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. Blinkers or monograms 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.

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.

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.

Other rules

A method for classifying RROs by the number of times they can fit into a single loop in a way that evenly divides the period has been discussed; by this logic, patterns such as the p160 oscillator in tlife, the p32 ("spinner") in DryLife, the p88 in B36ce7c/S23-y, and the natural p424 gun in Pedestrian Life could be classed as reflectorless rotating oscillators with a loopability of 1.

The multiplicity of a reflectorless rotating oscillator is the maximum number n of independent patterns that can orbit a single point, in a way that reduces the period of the combined oscillator by a factor of n.

Outer-totalistic rules

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

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:

• dmqwerty425 discovered a period-420 reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7 on Catagolue in November 2016.
• dani discovered a period-184 reflectorless rotating oscillator in B3/S23-a4i5i6ci in July 2017.
• Rhombic discovered a period-72 reflectorless rotating oscillator in B2e3-a4a/S1c23-aky in August 2017.
• Saka discovered a period-68 reflectorless rotating oscillator in B3-n4rtw5i/S23-n4q5i in August 2017.
• 2718281828 discovered a number of loopable RROs, with 2-, 3-, 4-, 5-, and even 6-fold loopability, in July 2018.
• 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.
• 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.

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

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.

• 

  Dean Hickerson's period-552 reflectorless rotating oscillator in the Larger than Life rule R7,C0,M1,S65..114,B65..95,NM.
  Catagolue: here

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  Dave Greene's 12-fold, period-46 version of Dean Hickerson's period-552 RRO.

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