Volatility

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The volatility of an oscillator is the size (in cells) of its rotor divided by the sum of the sizes of its rotor and its stator, that is, the proportion of cells (as a number from 0 to 1) involved in the oscillator which change state at some point during its period. The term "volatility" is due to Robert Wainwright.

The volatility of an oscillator may be very small (for example, a large oscillator with only two rotor cells such as boat tie spark coil which has a volatility of 0.08) or may be as large as 1 (where every cell in the oscillator is a rotor, such as pentadecathlon).

Oscillators with volatility 1

For many periods there are known oscillators with volatility 1 (also called pure rotor or statorless oscillators), such as Achim's p16, figure eight (p8), Kok's galaxy (p8), mazing (p4), pentadecathlon (p15), phoenix 1 (p2), p60 glider shuttle (p60), smiley (p8), and tumbler (p14). In these oscillators no cell is permanently on - that is, the stators are empty.

It is known that infinite families of volatility 1 oscillators can be formed either from glider shuttles such as relay,[1] or from periodic glider loops formed by no more than eight copies of a statorless 90-degree glider reflector.[2]

Periods for which non-trivial oscillators of volatility 1 are known are p2, p3, p4, p5, p6, p8, p12[note 1], p13, p14, p15, p16, p20, p22, p24[note 2], 25, p30, p32, p36[note 3], as well as 33n[3], 40 + 8n[4], 45 + 15n[1][2], 46n[5][6][2][7], 86n[8], 177n, and all periods given in the section Self-constructing circuitry below.

In particular, the smallest undecided period is 7, and prior to Dave Greene's infinite series of strictly volatile oscillators, the largest prime period for which such an oscillator was known was 13. Volatility 1 oscillators are known for all prime periods greater than or equal to 2918053.

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Statorless p3, a period-3 oscillator with volatility 1 (click above to open LifeViewer)
RLE: here Plaintext: here

Strict volatility

Strict volatility is a term that was suggested by Noam Elkies in August 1998 for the proportion of cells involved in a period n oscillator that themselves oscillate with period n. For prime n this is the same as the ordinary volatility. An oscillator is called strictly volatile if it has a strict volatility of 1, and trivial if it has a strict volatility of 0.

"Strictly volatile" is a very strong condition that excludes not only most patterns based on smaller-period oscillators (including glider shuttle or glider loop-based oscillators), but also some patterns that reflect or rotate on cell-centred axes or points partway through its period (such as twin bees shuttle and Gabriel's p138; related to kinetic symmetry). No infinite families of strictly volatile oscillators are known except for those based on self-constructing circuitry.

On December 2, 2022, Nico Brown published a script which can construct a strictly volatile oscillator for any period greater than or equal to 949.[9] The resulting patterns are extremely large, with the p1024 case (the only one run to completion so far) having a population of over 594 million cells. Further modifications to the script allowed for periods 943, 945, 946, 947, and 948 to be constructed.[10] Below this, the only periods for which strictly volatile oscillators are known are p2, p3, p4, p5, p6, p8, p13, p15, p16, p22, p25, p30, p33[note 4], p86, and p177.

Self-constructing circuitry

In November 2018, Dave Greene established using self-constructing circuitry that strictly volatile oscillators exist for all periods greater than or equal to 22178648.[11] The following month he reduced this to 3506916, and Goldtiger997 brought the minimum down to 3506910 a few days later by recompiling the same design.[12] There is also a known mechanism using this method for creating strictly volatile oscillators for periods that are not multiples of eight, between 2918053 and 3506909.[13]

See also

Notes

  1. A stabilization of the carnival shuttle using an unnamed statorless p4.
  2. A figure eight on statorless p3.
  3. A C4_4-symmetric traffic light hassler.
  4. An eightfold version of Jason's p33.

References

  1. 1.0 1.1 mniemiec (December 1, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  2. 2.0 2.1 2.2 FractalFusion (December 1, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  3. FractalFusion (December 10, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  4. Sokwe (December 1, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  5. Ivan Fomichev (January 4, 2015). Re: Thread for your unsure discoveries (discussion thread) at the ConwayLife.com forums
  6. James Pascua (August 31, 2020). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  7. Martin Grant (December 6, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  8. Matthias Merzenich (September 23, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  9. Nico Brown (December 2, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  10. Nico Brown (December 3, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  11. Dave Greene (November 21, 2018). Re: Self-Constructing Spaceship Challenges (discussion thread) at the ConwayLife.com forums
  12. Goldtiger997 (December 5, 2018). Re: Self-Constructing Spaceship Challenges (discussion thread) at the ConwayLife.com forums
  13. Chris Cain (November 30, 2018). Re: Self-Constructing Spaceship Challenges (discussion thread) at the ConwayLife.com forums

External links