Reflectorless rotating oscillator

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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 can be proven to exist (see universal constructor), but none have been explicitly constructed in Life. One of the isotropic-rule RROs listed below could be simulated using Adam P. Goucher's 0E0P metacell, and the result would closely resemble a standard single-loop RRO. However, 0E0P metacells have an unalterable orientation, so the pattern after N generations would never exactly match a rotated copy of the original until the pattern returns to its original configuration.

It's possible to use 0E0P metacells to create rotationally symmetric patterns that meet the technical specifications of an RRO, but only by arranging multiple separate meta-RRO loops in a symmetrical arrangement, with 0E0P metacells in a different orientation in each loop.

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.


Outer-totalistic rules

There is only one known 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.

Non-totalistic rules

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

  • dmqwerty425 discovered a period-420 reflectorless rotating oscillator in B2i34ik7/S23-a4ikn5j7 on Catagolue in November 2016.[1]
  • Daniel R. Collazo 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]

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.[6]


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;[7] by this logic, patterns such as the p160 oscillator in tlife, 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.


  1. Soup search results in rules other than Conway's Life (discussion thread) at the forums
  2. Thread for basic questions (discussion thread) at the forums
  3. Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the forums
  4. Thread for Your Accidental Discoveries that Aren't in CGOL (discussion thread) at the forums
  5. Reflectorless Rotating Oscillators (RRO) (discussion thread) at the forums
  6. Abstract Art (discussion thread) at the forums

External links