# 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.

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. 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.

## Other rules

### Outer-totalistic rules

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.

### Isotropic non-totalistic rules

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.
^{[1]} - dani discovered a period-184 reflectorless rotating oscillator in B3/S23-a4i5i6ci in July 2017.
^{[2]} - Rhombic discovered a period-72 reflectorless rotating oscillator in B2e3-a4a/S1c23-aky in August 2017.
^{[3]} - Saka discovered a period-68 reflectorless rotating oscillator in B3-n4rtw5i/S23-n4q5i in August 2017.
^{[4]} - 2718281828 discovered a number of loopable RROs, with 2-, 3-, 4-, 5-, and even 6-fold loopability, in July 2018.
^{[5]} - 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.
^{[6]} - 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.
^{[6]}

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.^{[7]}

## Classification & Multiplicity

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;^{[8]} 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.

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*.

## References

- ↑ Aidan F. Pierce (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
- ↑
^{6.0}^{6.1}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
- ↑ muzik (July 22, 2017). Re: Thread for basic non-CGOL questions (discussion thread) at the ConwayLife.com forums

## External links

- Multiplicity at the Life Lexicon

- Rotary oscillators? (discussion thread) at the ConwayLife.com forums