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 oscillator as well.^{[1]} 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 under period 50 include:
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. This one is unix on Rich's p16 , a p48 oscillator that was formerly the smallest.
x = 23, y = 17, rule = B3/S23
14bo3bo$13bobobobo$14b2ob2o2$11bo9bo$10bobo7bobo$10bo3b2ob2o3bo$11bo3b
obo3bo$12b2obobob2o$5b2o7bo3bo$5b2o3$5b3o$2o2bob2o$2o2b2o$4b2o!
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unix on Rich's p16 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
The example below was named "Obnoxious p230" in Dean Hickerson's oscillator collection.^{[note 2]} 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.
x = 33, y = 26, rule = B3/S23
9b2o$5b2o2bo$4bo2bobo$2obobobob2o$ob2obobobo$11bo$b4obob3obo$o4b3o4bo$
b3ob3ob3o$3b7o5$4b3o$19bo3b3o$18bobo2b5o3b2o$18bo3b2o3b2o2b2o$19bo2bo
3b2o$20b3o3bo2$20b3o3bo$19bo2bo3b2o$4b2o12bo3b2o3b2o$4b2o12bobo2b5o$
19bo3b3o!
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Obnoxious p230 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : 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) or 4 (overall 68) 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.
x = 25, y = 20, rule = B3/S23
11b2o5bo$11bo4b2obo$6bob2obo2b2o3bo$6b2obob2o3bo2bob2o$9bo5bo2b2obo2bo
$o8b2ob2o3bo5b2o$3o10bobob2o$3bo5b2obo2b2o3bo$2bo6b2obobo3b3o$3bo9bo2b
2o$6b2o6b2o2bo$6b2o9b2o2$7b2o$7b2o$11bo$4b2o6bo$5bo5bo$2b3o7b3o$2bo11b
o!
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The period-51 oscillator mentioned above, 90P51 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
x = 22, y = 20, rule = B3/S23
2bo11bo$2b3o7b3o$5bo5bo$4b2o5b2o2$7b2o$7b2o$6bobo$6b2o$6b2o$13bo$2b2o
5b2obobo2b2o$3bo5b2obo2bo2bo$3o10bob2obob2o$o8b2ob2o4bo2bo$9bo5b3obo$
10b3o5bo$13bob3o$10b2obobo$10bobo!
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The period-68 oscillator mentioned above, 72P68 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
11b2o$6b2o4bo$4b3o3bobob2o$bobo4bob2obobo$b2ob4obo3bo$4bo4bo3bo$4bob2o
bobob3o$3b2o2bobob2o3bo$5bo2b2o4b2o$b3o2b3o2b3o$obo3bobo2b2o$o3b2o2bo
5b2o$b3o4bob4o2bo$3bo5bo5b2o$11bo$10b2o!
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A period-13 component (53P13 ) acting with a period-2 component; this is 87P26 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
x = 26, y = 26, rule = B3/S23
9b2o$9bo$2o4b2obo$o2b2o3bo$b2o2$b2o$obobo3b2o$o3b2obo2bo$b4o10b2o$10bo
4bo$3b2o3bo3bo3b3o$3bo5bob5o2bo$5bo10bo$4b2o3b5o2b2o$8bo5bobo2bo$4b2o
3b2o2b2o3b2o$4bo2bobo5bo$6b2o2b4obo$7bo$5bo2b5obo9b2o$5b3o3bobo5b3o2b
2o$8bo10bo$7b2o9bobo$14b2ob2o$14b2o3bo!
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LCM(14,16)=112;^{[note 3]} 116P112 (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
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. Some examples are below:
x = 49, y = 28, rule = B3/S23
3b2o$3bo$5bo$4b4o$3bo4bo$3b5obo$b2o6bo7b2o$o2bobobob2o6bobo4bo12bo$2ob
obobo2bo6bobo5bo9bobo$3bo2bob2o8b2o3b2o3b2o3b2o$3b2obo3bo16bo2bo2b2o
12b2o$5bobob2o17b2o3b2o12b2o$5bobobo25bobo$6bo2bo27bo$7b2o$14b8o$12b3o
b4ob3o$11bo12bo$12b7o2b2obo$22b2o$14bob5o$13bobo5b3o$13bobo3b2o3bo$12b
2ob2obobob2o$19bo2bo$20bobo$18bobob2o$18b2o!
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Beehive hassler; LCM(18,30) = 90^{[4]} (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
x = 37, y = 26, rule = B3/S23
2o$2o5b2o$6bobo$6bo4bo$7b2o3bo$8bo2bo$10bo$6bo$5bo2bo$4bo3b2o$5bo4bo
14bo$8bobo12bobo$8b2o5b2o4b2o12b2o$15b2o4b2o12b2o$21b2o$23bobo$25bo2$
13b2o$12bo2bo$13bobo$14bo$10b2o7b2o$11bo7bo$8b3o9b3o$8bo13bo!
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Loaf flipper; LCM(40,30) = 120^{[5]} (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
x = 59, y = 55, rule = B3/S23
23b2o$23bobob2o$25bobo$24bo2bo$17b2ob2obobob2o$18bobo3b2o3bo$18bobo5b
3o$19bob5o$27b2o$17b7o2b2obo$16bo12bo$17b3ob4ob3o$19b8o4$22b2o$10b2o
10b2o$10bo7bo2bo$11b2o7b2o$8b4o6b4o$8b2o7b2o$8bo2bo7bo15b4o$6b2o10b2o
14b6o$6b2o25b8o$32b2o6b2o$33b8o$34b6o$3b8o24b4o$b3ob4ob3o$o12bo6b2o$ob
2o2b7o6bo2bo$b2o16bobo25b2o6b2o$4b5obo9bo25bo2bo4bo2bo$b3o5bobo33b6o2b
6o$o3b2o3bobo23bobo8bo2bo4bo2bo$b2obobob2ob2o22bobo9b2o6b2o$2bo2bo26bo
2bo2bo$2bobo27bo4bo$b2obobo24bo2bo2bo$5b2o6b2o6b2o9bobo$12bo2bo4bo2bo
8bobo$11b6o2b6o$12bo2bo4bo2bo$13b2o6b2o24b2obo$47b2ob3o$53bo$47b2ob3o$
31b4o13bobo$30b6o12bobo$29b8o12bo$28b2o6b2o$29b8o$30b6o$31b4o!
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Loaf flipper using a different reaction from the other two in this gallery; LCM(60,48) = 240^{[6]} (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
x = 49, y = 32, rule = B3/S23
b2o$b2o$20b2o10b2o$19bo2bo8bo2bo$2obo15b3o10b3o$bobo18b10o8bo$2bo18bo
2b6o2bo5b5o$b2o4b2o12b2o2b4o2b2o2bobo5bo$b2o4b2o26b2obobobobo2bo$b2o7b
obo25bo3bob4o$10bo2bo24bobobo$10bo2bo25bob7o$11b2o35bo$27b3o12bo3bobo$
27bo2bo7bo3b2o2b2o$31bo6bo4bo$27bo2bo7bo$27b3o11bobo$42bo2$b2o$b2o4b2o
$7b2o2$ob2o15b2o$obo15bo2bo$bo16bobo$b2o16bo$b2o10b2o7b2o$b2o11bo7bo$
11b3o9b3o$11bo13bo!
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Loaf flipper using the same reaction as the p120; LCM(76,72) = 1368^{[5]} (click above to open LifeViewer )RLE : here Plaintext : here Catagolue : here
See also
Notes
References