Trivial

A period-n oscillator is considered trivial if none of its cells oscillate at the full period, i.e. period n. By definition, an oscillator is trivial if and only if its strict volatility is precisely 0. For a trivial oscillator, it is necessary that the period n be a composite number that is not a prime power.

Most known trivial oscillators consist of two oscillators (possibly fused together) of periods k and m, which together oscillate at a greater period which is the least common multiple of k and m. Trivial oscillators are not generally considered interesting, except in special cases like welds or when the rotors touch.

Sometimes a trivial oscillator is useful for other purposes, such as a huge welded oscillator with p4 and p13 components in the p13-assisted period-52 glider gun. In this context, its triviality does not matter.

The definition of trivial has been questioned using the second example below. It has been suggested that the definition be modified as follows: An oscillator is considered trivial if no cell oscillates at the full period and none of the rotors interact with each other.

x = 31, y = 39, rule = B3/S23 5b2o$5b2o3$8b2o$9b2o8b2o$19b2o2$10bobo$9bo3bo$8bo3bo$7bo3bo$8bobo2$2o$ 2o8b2o$11b2o2$29b2o$14b2o13bo$14b2o11bobo$21b2o4b2o$20b2obo$21bob2o$ 22b2o$b3o4bob2o$5b3obo7b3o$6b2o8bo3bo$4b2ob2o7bo3bo$5bobobo6bob2o7b2o$ 4b2ob2o18bobo$4bo6b2o16bo$11b2o16b2o3$25b2o$25bobo$27bo$27b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 60 THUMBSIZE 2 ZOOM 8 HEIGHT 400 ]] Merzenich's p64 and a capped period-48 glider gun share a block. Despite not interacting, this oscillator is considered non-trivial by the above definition, since three cells oscillate at the full period of 192 (click above to open LifeViewer)

x = 5, y = 5, rule = B2kn3-ekr/S23-y 2bo$2o$2bobo$bobo$3bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 60 THUMBSIZE 2 ]] A period-6 oscillator in B2kn3-ekr/S23-y that is trivial by the above definition, as all cells oscillate at period 2 or 3, but the oscillator cannot survive without both parts (click above to open LifeViewer)

x = 27, y = 18, rule = B3/S23 11b2o$11bo6b2o$8b2obo6bo2bo$8bobob2ob2obob2o2bo$2bo7bo2bobo2bo2bobobo $2o2bo4b2o2bobobob2o2bo2bo$4bo9b2obobob2ob2o$2bo3b2obob3o3bo4b2o$11bo b2o8bo$bobobo2b3o2b2ob3obobob2o$bobo7bo10bobo$bo12b2ob2o3bobo$8bob5o3b o4bo$8b2o3bo2bo2bo$11b4ob3o$8b3o4bo$8bo2bo2bo2b2o$10b2o3b2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ]] Another period-6 oscillator in Life that is trivial by the above definition, as all cells oscillate at period 2 or 3, but the oscillator cannot survive without both parts, similar to the one above. (click above to open LifeViewer)

Gallery

This gallery includes trivial oscillators in Conway's Life where the rotors touch/interact.

x = 27, y = 18, rule = B3/S23 11b2o$11bo6b2o$8b2obo6bo2bo$8bobob2ob2obob2o2bo$2bo7bo2bobo2bo2bobobo $2o2bo4b2o2bobobob2o2bo2bo$4bo9b2obobob2ob2o$2bo3b2obob3o3bo4b2o$11bo b2o8bo$bobobo2b3o2b2ob3obobob2o$bobo7bo10bobo$bo12b2ob2o3bobo$8bob5o3b o4bo$8b2o3bo2bo2bo$11b4ob3o$8b3o4bo$8bo2bo2bo2b2o$10b2o3b2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ZOOM 12 ]]

Trivial p6
(click above to open LifeViewer)

x = 49, y = 54, rule = B3/S23 5b2o$5bo15b2o5b2o4b2o$6bo14bo5bo2bo4bo$3b4o8b2ob2obo5b2obo4bob2o3b2obo $3bo12bobo2b2o7b2o2b2obobo2bob2o2bo$6b3o6bo4bo2bob4o4bo3bo2bobo3b3o$6b o2bo6b5o2bobo3b3o3b2obobo2b2o$8b2o8bobo2b2o3bo7bo4bobob2o$2o12b3o6b2o 2b2obo3bo4b2o2bobobo$obo2b3o5bo3bo2b3o4bobobobo3b2o3bobo3bo$9b3o2bo2bo b2obo2bob2o5bo3bo8b2o$o2b2obobob3obo4b2ob5obo2b2obob5o3bo$bo5bo5bobo8b o6bo2bobo4bo2b5o$bo4bo3b2obo18bobo2b4o7bo$6bo4bo5bo11b3o2bo6b5o$12bobo 3b2o9bo4b2o3bo2bo2bo$4b2o3b2o4b3o2b2o17b2o$3bo3bob5obob3obo3b2o$3b2obo 8bo4b2o3bo2bo$4bob2ob7o3b2ob2obob2o$4bo2bobo12b2obo$5b2o5b2o4bobo4bo$ 12bo7b5ob2o$9b2obo3b2o3bo2bobo$10bob2obo4bo2bo2bo$9bo3bobo2bo3b2obo$ 10b3o2bobo2bo2bob2o$12bobo2bo3bobo2bo$14b4o2b3obo$13bo3b2o5b2o$12bo2bo 6bo$13b2obo6b3o$14bo6b2o3bo$14bob2o4bobo2bo$11b2obo4bob2ob4o$12bob2obo bo3bo$12bo4b4obo2bo$13b3o6bob2o$15b2o3bobo$18bo2b2o$15b4obo2b2o$15bo2b o2b2o2bo$16bobo4bobo$15b2obo2b3ob2o$15bo2b2obo2bo$13bobob2o2b2obo$13b 2o2b2o4bo$17b2o2b2o$18bo2bo$19bobo$18b2ob2o$19bobo$19bobo$20bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 5 THUMBSIZE 2 ZOOM 5 ]]

Trivial p10
(click above to open LifeViewer)

x = 84, y = 72, rule = B3/S23 9$24bob2ob2o$24b2ob2o2bo$30bo$24b5ob2o$23bo4b2o3b2o$23b4o4b2obo$32bo$ 25bo6bo5b2o18b2ob2o$25b7ob2o3bobo2bo11bo3bobo$33bobo2b7o9bo3bo2bo$27b 3o3bob4o4b2o10bo2bobo$34b2o2bobobo7b6ob2ob2o$26bo2bo6b3obob2o2b2o6bo4b o2bo$23b4o2bo10bobo2b2o3bo10b2o$23b3ob6o2bob3o2bo5bobobo2bo$30bobo4bo 4b2o10b2o$31bo3b2o$28bobo4b2obobo$28b3obo4bob2o$29bobobobobo$29bobob2o 2bo$26b2o3bobobobob2o$27bo3b5ob2obo$27bob2obobobo$28b2obo6b2o$33bobob 2o$28b2o10bo$30bob2o3b4o$33b5o$15b2o11b2obobob6o$12bob4o10b2obo5bo2bo$ 11bo15bobob2o4bo$10bo3bo2bobo12bo5b2o$11b2o6b2o6bobo2bo3bo$4b2o13b2o8b 2ob2o2bo2bo$5bo2bo6bo4b2o4b2ob4o3bobobo$5bobob2obobob2o11bobo2b3ob2o$ 3b2o2bobo2bobo17bo2bobo$4bobo4bo5b2o2b2o2b2o2bobob3obo$4bo2b2o2bob2o3b 2obo4bo2b2o5bo$3b2obo2bobobo5bob5o3b2o$4bob2obobobo4b2o2bo2bo3b2o15b2o $4bobo2bo2bo6bo2bo2bob2obo13bo2bo$5bo2b2o6b3obo5b2obo2bo2b2ob2o4b2o$ 14bo4bo7bobob2ob2o3bo$12b3ob4o7bo2bo3b2ob2o3b4o$12bobobobo12bob3o5bobo 2bobo$11b3obo3bo8b4obo3b2o3bo2bob2o$11bo6bo14b2obobo2bo2b2o$12b2obo18b obo6bobo$13bob2o18bob8o$13bo23bo$12b2o23bob2o2b4o$38bobo2bo2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ZOOM 5 ]]

Trivial p12
(click above to open LifeViewer)

x = 66, y = 54, rule = B3/S23 11bo2bo7b2o$11bo2bo7bo15b2o5b2o4b2o$8bob2o2b2o7bo14bo5bo2bo4bo$7bobo5b 3o2b4o8b2ob2obo5b2obo4bob2o3b2obo$8bo11bo12bobo2b2o7b2o2b2obobo2bob2o 2bo$b2o20b3o6bo4bo2bob4o4bo3bo2bobo3b3o$2bo2b2o5b2o2b3o4bo2bo6b5o2bobo 3b3o3b2obobo2b2o$2bobob3o2bobo11b2o8bobo2b2o3bo7bo4bobob2o$2o2bobo2bo 2b2obo15b3o6b2o2b2obo3bo4b2o2bobobo$bobo3b2o2bo2b6o2b3o5bo3bo2b3o4bobo bobo3b2o3bobo3bo$bob3o2bob2o2bobo9b3o2bo2bob2obo2bob2o5bo3bo8b2o$2obo 2bobobo7bob2obobob3obo4b2ob5obo2b2obob5o3bo$bob2obobobo4b2o7bo5bobo8bo 6bo2bobo4bo2b5o$bobo2bo2bo7b2o4bo3b2obo18bobo2b4o7bo$2bo2b2o7b2obo5bo 4bo5bo11b3o2bo6b5o$11bo5bo11bobo3b2o9bo4b2o3bo2bo2bo$9bobobo2bo4b2o3b 2o4b3o2b2o17b2o$13bo6bo3bob5obob3obo3b2o$8bob3ob3o3b2obo8bo4b2o3bo2bo$ 8bo12bob2ob7o3b2ob2obob2o$9b2ob2o7bo2bobo12b2obo$10bob2o8b2o5b2o4bobo 4bo$10bo18bo7b5ob2o$9b2o15b2obo3b2o3bo2bobo$27bob2obo4bo2bo2bo$26bo3bo bo2bo3b2obo$27b3o2bobo2bo2bob2o$29bobo2bo3bobo2bo$31b4o2b3obo$30bo3b2o 5b2o$29bo2bo6bo$30b2obo6b3o$31bo6b2o3bo$31bob2o4bobo2bo$28b2obo4bob2ob 4o$29bob2obobo3bo$29bo4b4obo2bo$30b3o6bob2o$32b2o3bobo$35bo2b2o$32b4ob o2b2o$32bo2bo2b2o2bo$33bobo4bobo$32b2obo2b3ob2o$32bo2b2obo2bo$30bobob 2o2b2obo$30b2o2b2o4bo$34b2o2b2o$35bo2bo$36bobo$35b2ob2o$36bobo$36bobo$ 37bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ZOOM 5 ]]

Trivial p15
(click above to open LifeViewer)

x = 97, y = 99, rule = B3/S23 7$50bob2ob2o$50b2ob2o2bo$56bo$50b5ob2o$49bo4b2o3b2o$49b4o4b2obo$58bo$ 51bo6bo5b2o18b2ob2o$51b7ob2o3bobo2bo11bo3bobo$59bobo2b7o9bo3bo2bo$53b 3o3bob4o4b2o10bo2bobo$60b2o2bobobo7b6ob2ob2o$52bo2bo6b3obob2o2b2o6bo4b o2bo$49b4o2bo10bobo2b2o3bo10b2o$49b3ob6o2bob3o2bo5bobobo2bo$56bobo4bo 4b2o10b2o$57bo3b2o$54bobo4b2obobo$54b3obo4bob2o$55bobobobobo$55bobob2o 2bo$52b2o3bobobobob2o$53bo3b5ob2obo$53bob2obobobo$54b2obo6b2o$59bobob 2o$54b2o10bo$42b2o12bob2o3b4o$42bo16b5o$45b4o5b2obobob6o$18b2o4b2o5b2o 11b3obo5b2obo5bo2bo$18bo4bo2bo5bo11b3o6bobob2o4bo$8bob2o3b2obo4bob2o5b ob2ob2o19bo5b2o$5bo2b2obo2bobob2o2b2o7b2o2bobo15bobo2bo3bo$5b3o3bobo2b o2bo5b4obo2bobo2bo7b2o7b2ob2o2bo2bo$9b2o2b3o2bo3b3o3bobo2bob3o6bo2bo4b 2ob4o3bobobo$7b2obobo5bo3bo2b2obo2bo12bo2bo7bobo2b3ob2o$6bo3bob3o7b2o 2b4o7b3o5bobo10bo2bobo$5bob2obo2bo5b2o4bo2b3ob2o2bo3bo6b2o2b2o2bobob3o bo$5b2obobo6bo7bo2bobo5bob2o4b2o6bo2b2o5bo$9b2o2b2o2bobobo2bo3bo5bo3b 2o8b4o3b2o$5b5o2bo4bobo2bo11bo3bob5o3bo2bo3b2o15b2o$5bo7b4o2bobo17b2ob ob3obo2bob2obo13bo2bo$8b5o6bo2b3o10bo3bo4bo2bo4b2obo2bo2b2ob2o4b2o$8bo 2bo2bo3b2o4bo8b3o3bob2o10bobob2ob2o3bo$13b2o17bo3bob2obo3b2ob2o3bo2bo 3b2ob2o3b4o$27b2o6bo2b2o3b3o4bo6bob3o5bobo2bobo$25bo2bo4bo4bo8bob2o3b 4obo3b2o3bo2bob2o$25b2obob5o3b7ob2obo9b2obobo2bo2b2o$28bob2o12bobo2bo 10bobo6bobo$28bo6bo4b2o5b2o12bob8o$26b2ob2ob4o5bo21bo$27bobo2b2ob3o3bo b2o18bob2o2b4o$27bo2bob2o4bob2obo20bobo2bo2bo$28bobobo3bobobo3bo$27b2o bob3obobo2b3o$27bo2bo2b4o2bobo$29bob2obo2b3o$28b2obo3b2o3bo$31bo3b4o2b o$28b3obo6b2o$27bo3b3o5bo$26bo2bo3bo2b2obo$26b4ob3o2bo2bob2o$30bo3bobo b2obo$28bo2bob2o6bo$28b2obobob2ob3o$31bo3bob2o$31bo$29b2o2bobob2o$28bo 2b3obo2bo$28bobo4bobo$27b2ob2obobob2o$29bo4bo3bo$29bob4o3bobo$30bo3bo 4b2o$31b2obo$32bobo$32bo3bo$31b2ob2o$32bobo$32bobo$33bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ZOOM 3 ]]

Trivial p20
(click above to open LifeViewer)

x = 24, y = 16, rule = B3/S23 22bo$21bobo$3b3o2bo13bo$4b2obo9bo$4b3o4bo3bob2obob2o$o3bobo8bob4o$4o2b o6bo$3o15b3o$2b3o2bo11b5o$bo4bobo3b2o8bo$o6bo2b2o2$8bo$2bo5bo$7bo$4bo 2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 6 THUMBSIZE 2 ]]

Trivial p80. The LCM cell is p40, 34P20 on Rob's p16. Notably, this oscillator is also statorless.
(click above to open LifeViewer)

See also

• Non-trivial
• Omniperiodic
• Trivial oscillators (category)

External links

• Trivial at the Life Lexicon
• Triviality Tester (discussion thread) at the ConwayLife.com forums