# Difference between revisions of "LCM oscillator"

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==Completions of partials== | ==Completions of partials== | ||

Occasionally, a [[hassler]] partial can be completed with an LCM reaction but not with the base period. A p345 | Occasionally, a [[hassler]] partial can be completed with an LCM reaction but not with the base period. A p345 using [[Karel's p15]] is an example, with LCM(69,15)=345. The second p52 on the [[Lumps of muck hasslers#p52|lumps of muck hasslers]] page was an example for one day; p104 ([[figure eight]]) and p152 ([[unix]]) completions were known one day before the toad that kept it at p52 was discovered. | ||

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{{gallery item|{{EmbedViewer | {{gallery item|{{EmbedViewer |

## Revision as of 06:06, 12 March 2024

The **least common multiple** (**LCM**) of two nonzero integers *a* and *b*, usually denoted by lcm(*a*, *b*), is the smallest positive integer that is divisible by both *a* and *b*.^{[1]}

A **least-common-multiple oscillator**, often abbreviated to **LCM oscillator** or **LCM**, is a non-trivial^{[note 1]} oscillator that has two parts that interact only when each part is in a particular phase, which results in an overall period of the least common multiple of the periods of the smaller parts. There are several types of LCM oscillators.

## Dying sparks

The most common type of LCM oscillator is one where dying sparks interact. They are thus named **spark-coupled oscillators** as well.^{[2]} These are often the smallest of their period by population. At least one of the oscillators needs to have a two-cell (or greater) spark; if both are dot sparkers, there will not be enough cells for a birth between them. Those that are SKOPs and period 50 or below include:

- Mold on fire-spitting; 3 × 4 = 12; 31 cells
- Mold on fumarole; 4 × 5 = 20; 30 cells
- Caterer on figure eight; 3 × 8 = 24; 24 cells
- Mold on 34P14 shuttle; LCM(4,14) = 28; 46 cells
- Caterer on rattlesnake; 3 × 11 = 33; 46 cells
- Caterer on Beluchenko's p13; 3 × 13 = 39; 46 cells
- Unix on 34P14 shuttle; LCM(6,14) = 42; 45 cells
- Mold on rattlesnake; 4 × 11 = 44; 44 cells
- Unix on Charity's p16; 3 × 16 = 48; 44 cells
- 24P10 on 30P25; LCM(10,25) = 50; 58 cells

Note that if the two periods are coprime, the resulting period will be their product, but this is not the case if they aren't coprime.

Below is a typical example of an LCM oscillator. This one is unix on Rich's p16, a p48 oscillator that was formerly the smallest.

unix on Rich's p16 (click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

The example below was named "Obnoxious p230" in Dean Hickerson's oscillator collection. It was found by Bill Gosper on May 28, 1992.^{[3]} It has a period of 5 × 46 = 230. What makes this one slightly different is that two of the five p46 cycles create a temporary object that is then quickly destroyed.

Obnoxious p230 (click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

## Single objects

Honey thieves (period 17) has no accessible sparks; however, there are ways of replacing one of the fishhooks with a period-3 (overall 51), 4 (overall 68) or 5 (overall 85) component. Other periods with billiard tables can do the same. To qualify as non-trivial, there needs to be at least one cell that oscillates at the full period; all of those below do.

The period-51 oscillator mentioned above, 90P51 (click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

The period-68 oscillator mentioned above, 72P68 (click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

A period-13 component (53P13) acting with a period-2 component; this is 87P26 (click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

LCM(14,16) = 112;^{[note 2]} 116P112(click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

A period 19 component (Cribbage) interacting with p5 billiard table by Matthias Merzenich to form 141P95^{[5]}(click above to open LifeViewer) Catagolue: here |

## Hasslers

*See also: Hassle*

There are some objects that can be pushed with more than one oscillator. In many cases, the reaction fails in some generations, so examples with higher greatest common factors are more likely to work, as they skip over the problematic generations. Sometimes, phase offsets are allowed. Some examples are shown below:

Beehive hassler; LCM(18,30) = 90^{[6]}(click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

Loaf flipper; LCM(40,30) = 120^{[7]}(click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

Loaf flipper using a different reaction from the other two in this gallery; LCM(60,48) = 240^{[8]}(click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

Loaf flipper using the same reaction as the p120; LCM(76,72) = 1368^{[7]}(click above to open LifeViewer) RLE: here Plaintext: hereCatagolue: here |

## Completions of partials

Occasionally, a hassler partial can be completed with an LCM reaction but not with the base period. A p345 using Karel's p15 is an example, with LCM(69,15)=345. The second p52 on the lumps of muck hasslers page was an example for one day; p104 (figure eight) and p152 (unix) completions were known one day before the toad that kept it at p52 was discovered.

A period-345 LCM oscillator. The base reaction is p69. As 69 is a multiple of 3, Karel's p15 can push the beehive back in generations that are multiples of 3, avoiding the two in a row that cause it to fail. No non-LCM completion of this p69 is known. (click above to open LifeViewer) |

p1064 70 × 70 emu^{[citation needed]} made with 4 copies of the R133 conduit. Despite the fact that the four Herschels reappear in the same locations every 133 ticks, there is a Coe's p8 that results in an LCM oscillator.(click above to open LifeViewer) |

## See also

- Least-common-multiple oscillators (category)

## Notes

- ↑ Trivial oscillators are typically not counted in this definition.
- ↑ Found by Noam Elkies on September 14, 2014
^{[4]}

## References

- ↑ Least common multiple at Wikipedia
- ↑ User:Tropylium/Spark-coupled oscillator
- ↑ Dean Hickerson's oscillator stamp collection. Retrieved on November 14, 2021.
- ↑ Matthias Merzenich (April 18, 2018). "osc-supplement pattern collection". Retrieved on November 14, 2021.
- ↑ Matthias Merzenich (July 14, 2023). Re: Smallest Known Oscillators to p106 (and Beyond) (discussion thread) at the ConwayLife.com forums
- ↑ 83bismuth38 (November 1, 2021). Message in #cgol on the Conwaylife Lounge Discord server
- ↑
^{7.0}^{7.1}David Raucci (October 31, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums - ↑ David Raucci (October 26, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums