LCM oscillator

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.

A least-common-multiple oscillator, often abbreviated to LCM oscillator or LCM, is a non-trivial 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. 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:

• Mold on fire-spitting; 3 × 4 = 12
• Mold on fumarole; 4 × 5 = 20
• Mold on 34P14 shuttle; LCM(4,14) = 28
• Caterer on rattlesnake; 3 × 11 = 33
• Caterer on Beluchenko's p13; 3 × 13 = 39
• Unix on 34P14 shuttle; LCM(6,14) = 42
• Mold on rattlesnake; 4 × 11 = 44
• Pentadecathlon on thumb 1; LCM(15,9) = 45
• 30P3 on Rob's p16; 3 × 16 = 48

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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 12 GPS 12 ]]

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. It was found by Bill Gosper on May 28, 1992. 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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 10 GPS 60 ]]

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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 GPS 26 ]]

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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 GPS 34 ]]

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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 GPS 13 ]]

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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 10 GPS 56 ]]

LCM(14,16) = 112; 116P112 (click above to open LifeViewer) RLE: here Plaintext: here 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. 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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 GPS 45 ]]

Beehive hassler; LCM(18,30) = 90 (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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 GPS 60 ]]

Loaf flipper; LCM(40,30) = 120 (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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THUMBSIZE 2 ZOOM 8 GPS 60 ]]

Loaf flipper using a different reaction from the other two in this gallery; LCM(60,48) = 240 (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! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THUMBSIZE 2 ZOOM 8 GPS 60 ]]

Loaf flipper using the same reaction as the p120; LCM(76,72) = 1368 (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here

See also

• Least-common-multiple oscillators (category)