OCA:Day & Night

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Day & Night
Day & Night rule
Rulestring 34678/3678
B3678/S34678
Rule integer 242120
Character Stable
Black/white reversal B3678/S34678
Strobing equivalent B01245/S0125

Day & Night is a Life-like cellular automaton in which cells survive from one generation to the next if they have 3, 4, 6, 7, or 8 neighbours, and are born if they have 3, 6, 7, or 8 neighbours.

Day & Night is the most well-known self-complementary rule; that is, if all grid cells have their on/off state exchanged, the history of the pattern is the inverse of the history of the original.

Nathan Thompson explored the rule starting in April 1997, and David Bell discussed the rule in detail the following November.

The rule is rich in spaceships, puffers, and oscillators.

On August 14, 2020, Peter Naszvadi constructed a Rule 110 unit cell in Day & Night, proving the rule Turing-complete.[1]

Oscillators

Day & Night is home to many different periods of oscillators. Here are the smallest known examples of each period up to 16:

x = 183, y = 12, rule = B3678/S34678 2o5bo28b2o9b2o55bo77bo$2o5b2o5b4o7bo8b4o9b2o10bobo20b2o5b2o11bob3o9bo 2bo9bobo7b2o9b3o28b2o$8bo6b2o7b3o7b6o6b5o7b6o6b5o6b3o5b3o10b5o9b3obo9b o8b2o10b2o28b2o$15b2o8bo10b4o7b4o7b6o7b3o7b2o7b2o6b2o2b4o10b5o8bo11bo 9bo3bo7b2ob2o4b2o$15b2o18bob2o9bo9bo2b2o23bo11b9o8b5o8b2ob2o9b2o11b2o 6b4o3b4o$14b4o19b2o22b3o21b3o11b4o2b2o7b5o12bo10b2o11b2o7b9o$60b3o9bo 11b5o10b2o12b5o12bo32b4o3b4o$70b2o13b3o25bob3o11bobo31b2o4b2ob2o$72bo 13bo27bo2bo$70bo10b2o7b2o$71b2o8b3o5b3o$70bo11b2o5b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
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Spaceships

Many periods of C/2 spaceships are known, but different reduced speeds are common as well. The S8 transition allows many of the spaceships in Day & Night to be stable.

Elementary

Speed Direction Smallest known Minimum # of cells
c/2 orthogonal 29P2H1V0 29
3c/7 orthogonal 584P7H3V0 584
2c/5 orthogonal 164P5H2V0 164
c/3 orthogonal 41P3H1V0 41
2c/7 orthogonal 290P7H2V0 290
c/4 orthogonal 32P4H1V0 32
c/5 orthogonal 40P5H1V0 40
c/6 orthogonal 63P6H1V0 63
c/7 orthogonal 104P7H1V0 104
c/8 orthogonal 190P8H1V0 190
c/9 orthogonal 212P9H1V0 212
(2,1)c/5 oblique 343P5H2V1 343
(2,1)c/6 oblique 532P6H2V1 532
c/3 diagonal 12P3H1V1(Butterfly) 12
c/4 diagonal 24P4H1V1 24
c/5 diagonal 83P5H1V1 83
c/6 diagonal 184P6H1V1 184
c/7 diagonal 48P7H1V1 48

A collection of these ships:

x = 385, y = 138, rule = B3678/S34678 b4ob2o12bo8bo15b3o10b3o14bo5bo10b4o6b4o7bo4bobo4bo8bo2bo5bo2bo15bo16bo 5bo14b2o14b3o2b3o13bo18b3o11b3o17b3o27b2o27b3o2b2o21bo$obo3b3o10b3o6b 3o12b3ob3o6b3ob3o9b5o3b5o8b2o3b2o3b2o8b2o3b3o3b2o8b2ob2o3b2ob2o14bobo 14bobo3bobo13b2o12bob2o4b2obo9b4o17b2obo9b3obo15b5o24bob3o24bo3b5o21b 3o$5ob4o8b4o6b4o10b3o2b4ob2ob4o2b3o8b4ob3ob4o8bo2b2o2b2o2bo9b2o2b3o2b 2o8b2o4bobo4b2o13b3o14b3o3b3o26b14o8b5o14b3o2b2o11b2o15b6o24b4o24b3o2b 3o22b4o$4b5o12b2o4b2o14b2o2b3ob4ob3o2b2o9bob2o5b2obo9b10o12b2o3b2o10b 3o2b5o2b3o13bobo33b2ob2ob2o9bo4b2o4bo9b5o13bobo4b2o10b2o17b5o23b2o26b 5ob2o25b2o$b2o2bo3bo7bo6b2o6bo9bob3o2bob4obo2b3obo9b2o7b2o11b8o14b2ob 2o12bobob5obobo31b2o5b2o9b2o2b2o2b2o8b2ob2o2b2ob2o9b4o13b4o2b4o10bo19b 3o21b3o2bo25bob2obob2o21bob4o$18b4o6b4o10b3o5b6o5b3o8bobo7bobo11bob2ob o15b5o55b2o7b2o9b8o11bobo2bobo10bobo13bo2bobo4b2o31bobo19bob2o27bob5o 2bo20bo3bob2o$20b2ob4ob2o12bo7bo4bo7bo7b2o11b2o11bo2bo34bo9bo31bob2o3b 2obo10b6o13b2o2b2o12b2o2b2o9bo2b6o3bo29bobo19bobobo28b2ob4obo16b2obo6b 2o$20b10o10b2o2bo6b4o6bo2b2o30b2o2b2o17bo41b3o15bobobobo10bob6obo10b8o 11bo3b2o9b3o3bobo34bo19bobobo27b6ob2o18bobob2o2bob3o$21b8o14bobo5bo2bo 5bobo31b3o4b3o57b3o15b2o3b2o12b6o13bob2obo10bo2b2o17bo2b2o52bobobo29bo b2ob4o17b2o2b2obo$18b2o2b6o2b2o11b2o16b2o30b3obo2bob3o54bo2bo2bo15bobo 15b4o15bo2bo11bo2bo3b2o16b2o51bobobo27bo2b3obo22bobo2bo$17b3o2b6o2b3o 9bo2b3o10b3o2bo28b5o4b5o52bob5obo32b4o31bo3b4o8bo6bobo50bobobo24b3ob3o 2bo25bo2bo$17bo2bob6obo2bo12b2obo8bob2o31b5o4b5o52b9o12b2obob2o13b4o 14bo4bo13bobo2bo8b3o4bo2b2o46b2obobo26b4obo29b2obo$17b4obob2obob4o8b8o 8b8o27bob10obo49b4ob5ob4o9b7o11b2ob2ob2o12b2o2b2o15bob2o9bob2o2bo50bob obo24b10o29b3o$15b3ob12ob3o7bob4o10b4obo30b10o51b2obob5obob2o10b5o13bo b2obo12bobo2bobo10b6obo8b2ob3o3bo49bobo23bob8o$17b4o3b2o3b4o9b2ob4o8b 4ob2o30b10o52b2o4bo4b2o9b3o3b3o10bo6bo9b5o2b5o10b5obo10bo4b3o46b2ob2o 23b11o$17b16o11bo2bo10bo2bo35bo2bo61bo13b3o7b3o11b2o14b3o2b3o11b9o13bo 2bo47bo25bob10o$20bo3b2o3bo16bo12bo38b2o77b5ob5o11b4o13b8o10bob5o16bob o46bo26bob9obo$23b4o21bo8bo39b4o78b7o14b2o11b3o3b2o3b3o7b7o15bob2o45b 2o26b11o$21b8o18b2o8b2o34b2o3b2o3b2o72b11o11bo2bo11bo2bo4bo2bo7bob6o 15bo2b4o38bo3bo25bo2b10o$21b8o19bo8bo35b3ob4ob3o76b3o15b4o11b2obob2obo b2o8b4o18bo5b2o36b4o28b2o2b6o$19bob8obo62b4ob2ob4o75b5o15b2o12b3ob4ob 3o6bob4obo18bob4o33b7o27bo2bo2b3ob2o$19bob8obo61bo3b6o3bo75b3o14b6o12b ob4obo10b5ob3o12b3o2bobo34b3o4bo26b5ob2obo$18b14o63b3o2b3o95bob2obo12b 8o11b8o11b4o38bob4o31b2obob2o$17b16o62bobo2bobo92b2o2b4o2b2o8b10o9b3ob ob2obo11b2o2b2o36b5o30bobobo$19b12o64bo6bo91bo2b8o2bo8b8o12bo5bo11bo4b o38b2o34b2obo$20b10o66bo4bo92b14o6b12o11b2o19bo39b2o$20b10o62bo3bo4bo 3bo89b12o7b2ob6ob2o12b2obo15b2obo38bo$19bob8obo61b3obo4bob3o88bob10obo 29b3o14b6o$22b6o64b3o8b3o89b12o30b3o13bo2b4o$20bob6obo64bo8bo91bo3bo2b o3bo11b4o16b3o9b2ob2ob2obo$21b8o62bo3bo6bo3bo110b6o14bob3o8bo3bo4b3o$ 19bo2b6o2bo61b2obo6bob2o92b6o14b4o30bo6bob2o$22b6o64b3obob2obob3o91b8o 11b2o4b2o13bo2bo10b2o3bo4bo$20bob6obo68b2o97bo2b2o2bo11bobo2bobo14bobo 12b2ob4o2bo$22b2o2b2o65bobobo2bobobo92bo2b2o2bo11b2ob2ob2o14bob2o9bobo 2bo2b2o$21bo6bo64b2o2bo2bo2b2o94bo2bo36bo13b2obo2bobo$21b3o2b3o64b5o2b 5o113bo2bo15b2o3bo10bo2bo4b3o$19bo2bob2obo2bo64b3o2b3o115bo2bo15b2o12b o2bobo4bobo$20bobo4bobo63bob8obo114b2o15b2o2b2o9b5o3b2o$17bob5o2b5obo 59bo4b4o4bo132b2o13b2obobobob2o$17b5obo2bob5o59bobo2b4o2bobo134bo13b2o 2bobob2o$16bob3o8b3obo58b5ob2ob5o131bo15b2o6bo$17b2ob2o6b2ob2o62b2o4b 2o133b2obob2o10b4o2b5o$16bo3bo8bo3bo59bo2bo4bo2bo132bo3b2o12bo3b5o$16b obo12bobo59b3o6b3o136bo11bob3ob3o$238b2o12bob2o6bo$16b2ob2o8b2ob2o204b 2o14b2o$17bo2b2o6b2o2bo205bo12b2o3bo$16bo2b3o6b3o2bo202b4o12bobo4b3o$ 18b5o4b5o204b4obo11b4o3b2o$16b6o6b6o201bob5o10bob2o$16b5o8b5o202b6o12b 3o$16bob4o6b4obo200b6o17b3o$18b2obo6bob2o202b5obo12bob6o$17b2o12b2o 200b4o17b2obob2o$17bo14bo200b7o15bob2o$16b3obo8bob3o199b5o15b2o2b4o$ 16b3o12b3o199b6o14bo5b2o$15b6o8b6o200b7o13b3ob2o$15b3obobo6bobob3o201b 6o12bob3o$16b3ob2o6b2ob3o203b4o15b2ob2o$15b3o5bo2bo5b3o203bobo15bob2o$ 15b3obo2b6o2bob3o202b3o20b2o$17bo2b10o2bo202b4o22bo$19b12o204b5o19bob 2o$20b10o204b6o21b2o$20bob6obo208b2o19b4o$20b10o206bo18b6obo$23bo2bo 208b3o14b2ob3ob2obo$21b8o209b2o12b2ob3obo$21b2ob2ob2o206b3obo13bob3obo $21bobo2bobo206b2ob3o11bob4o$23bo2bo207bo2b3o15bo$21b3o2b3o206b4o13b2o 2bo$20b2obo2bob2o203bo2b2o16bo2b2o$19b2obo4bob2o202b5o17b4o$19bob3o2b 3obo202b4o17b2ob2o$19b2obo4bob2o203bob2o15bo2bo$21b2o4b2o205b3o19bo$ 18bo12bo223b2o$19b2o8b2o202bo2b2o$235b2o2$237b2o$234bob3o$237b3o2$233b 2ob2obo$233bo4b3o$236bob3o$233b2o3b2o$233b3ob3o$237bobo$235b4o$233bo3b 3o$234bo$232b4o2b2o$234bobob3o$232bobo5b2o$237bo3bo$233bo2b3ob3o$235bo 4bo$234bo3b2obo$237b2o$236bob2obo$234b3ob2o$237b2o$234bobobobo$235b2ob 4o$237b2obo$235b3obo2$234b3o$236b2o$234b2ob3o$234bo4bo$236b3o$238bo$ 239b2o$237b2obo$237b2obo$237bo$234b6o$234b6o$233bob3o$234b7o$233bob5ob o$233bob6o$233b9o$235b5o$233b6o$233bo2b4o$233bo2b3obo$234b6o$234b5o$ 234b5o$232b2o2bo$234b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
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References

  1. Peter Naszvadi (August 14, 2020). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums

External links