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 ]]
(click above to open LifeViewer)

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 288P6H2V1 288
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 = 412, y = 102, rule = B3678/S34678 b4ob2o12bo8bo15b3o10b3o14bo5bo10b4o6b4o7bo4bobo4bo8bo2bo5bo2bo15bo16bo 5bo14b2o14b3o2b3o44bo14b3o11b3o17b3o27b2o27b3o2b2o21bo$obo3b3o10b3o6b 3o12b3ob3o6b3ob3o9b5o3b5o8b2o3b2o3b2o8b2o3b3o3b2o8b2ob2o3b2ob2o14bobo 14bobo3bobo13b2o12bob2o4b2obo40b4o13b2obo9b3obo15b5o24bob3o24bo3b5o21b 3o$5ob4o8b4o6b4o10b3o2b4ob2ob4o2b3o8b4ob3ob4o8bo2b2o2b2o2bo9b2o2b3o2b 2o8b2o4bobo4b2o13b3o14b3o3b3o26b14o39b5o10b3o2b2o11b2o15b6o24b4o24b3o 2b3o22b4o$4b5o12b2o4b2o14b2o2b3ob4ob3o2b2o9bob2o5b2obo9b10o12b2o3b2o 10b3o2b5o2b3o13bobo33b2ob2ob2o9bo4b2o4bo39bob5o8bobo4b2o10b2o17b5o23b 2o26b5ob2o25b2o$b2o2bo3bo7bo6b2o6bo9bob3o2bob4obo2b3obo9b2o7b2o11b8o 14b2ob2o12bobob5obobo31b2o5b2o9b2o2b2o2b2o8b2ob2o2b2ob2o39bob3obo7b4o 2b4o10bo19b3o21b3o2bo25bob2obob2o21bob4o$18b4o6b4o10b3o5b6o5b3o8bobo7b obo11bob2obo15b5o55b2o7b2o9b8o11bobo2bobo40bo2b5o5bo2bobo4b2o31bobo19b ob2o27bob5o2bo20bo3bob2o$20b2ob4ob2o12bo7bo4bo7bo7b2o11b2o11bo2bo34bo 9bo31bob2o3b2obo10b6o13b2o2b2o41b2obo9bo2b6o3bo29bobo19bobobo28b2ob4ob o16b2obo6b2o$20b10o10b2o2bo6b4o6bo2b2o30b2o2b2o17bo41b3o15bobobobo10bo b6obo10b8o39bob3obo7b3o3bobo34bo19bobobo27b6ob2o18bobob2o2bob3o$21b8o 14bobo5bo2bo5bobo31b3o4b3o57b3o15b2o3b2o12b6o13bob2obo41bo17bo2b2o52bo bobo29bob2ob4o17b2o2b2obo$18b2o2b6o2b2o11b2o16b2o30b3obo2bob3o54bo2bo 2bo15bobo15b4o15bo2bo42bob2o17b2o51bobobo27bo2b3obo22bobo2bo$17b3o2b6o 2b3o9bo2b3o10b3o2bo28b5o4b5o52bob5obo32b4o62b2o10bo6bobo50bobobo24b3ob 3o2bo25bo2bo$17bo2bob6obo2bo12b2obo8bob2o31b5o4b5o52b9o12b2obob2o13b4o 14bo4bo43b2o9b3o4bo2b2o46b2obobo26b4obo29b2obo$17b4obob2obob4o8b8o8b8o 27bob10obo49b4ob5ob4o9b7o11b2ob2ob2o12b2o2b2o43b2o10bob2o2bo50bobobo 24b10o29b3o$15b3ob12ob3o7bob4o10b4obo30b10o51b2obob5obob2o10b5o13bob2o bo12bobo2bobo41bobo9b2ob3o3bo49bobo23bob8o$17b4o3b2o3b4o9b2ob4o8b4ob2o 30b10o52b2o4bo4b2o9b3o3b3o10bo6bo9b5o2b5o39bo14bo4b3o46b2ob2o23b11o$ 17b16o11bo2bo10bo2bo35bo2bo61bo13b3o7b3o11b2o14b3o2b3o41b3o16bo2bo47bo 25bob10o$20bo3b2o3bo16bo12bo38b2o77b5ob5o11b4o13b8o41b2obo15bobo46bo 26bob9obo$23b4o21bo8bo39b4o78b7o14b2o11b3o3b2o3b3o38b2obo14bob2o45b2o 26b11o$21b8o18b2o8b2o34b2o3b2o3b2o72b11o11bo2bo11bo2bo4bo2bo34bo2bo2b 2o15bo2b4o38bo3bo25bo2b10o$21b8o19bo8bo35b3ob4ob3o76b3o15b4o11b2obob2o bob2o34bob3o18bo5b2o36b4o28b2o2b6o$19bob8obo62b4ob2ob4o75b5o15b2o12b3o b4ob3o34b5o20bob4o33b7o27bo2bo2b3ob2o$19bob8obo61bo3b6o3bo75b3o14b6o 12bob4obo36b3o19b3o2bobo34b3o4bo26b5ob2obo$18b14o63b3o2b3o95bob2obo12b 8o35b3obo17b4o38bob4o31b2obob2o$17b16o62bobo2bobo92b2o2b4o2b2o8b10o37b o19b2o2b2o36b5o30bobobo$19b12o64bo6bo91bo2b8o2bo8b8o33bob3o19bo4bo38b 2o34b2obo$20b10o66bo4bo92b14o6b12o31bo27bo39b2o$20b10o62bo3bo4bo3bo89b 12o7b2ob6ob2o31b2o25b2obo38bo$19bob8obo61b3obo4bob3o88bob10obo48bobo 22b6o$22b6o64b3o8b3o89b12o49b3o21bo2b4o$20bob6obo64bo8bo91bo3bo2bo3bo 11b4o32b7o16b2ob2ob2obo$21b8o62bo3bo6bo3bo110b6o30bob4o18bo3bo4b3o$19b o2b6o2bo61b2obo6bob2o92b6o14b4o32b3obo20bo6bob2o$22b6o64b3obob2obob3o 91b8o11b2o4b2o30b2o22b2o3bo4bo$20bob6obo68b2o97bo2b2o2bo11bobo2bobo27b 3ob3o22b2ob4o2bo$22b2o2b2o65bobobo2bobobo92bo2b2o2bo11b2ob2ob2o27bobob o22bobo2bo2b2o$21bo6bo64b2o2bo2bo2b2o94bo2bo48b4o25b2obo2bobo$21b3o2b 3o64b5o2b5o113bo2bo31bo26bo2bo4b3o$19bo2bob2obo2bo64b3o2b3o115bo2bo30b o25bo2bobo4bobo$20bobo4bobo63bob8obo114b2o29bo2bo24b5o3b2o$17bob5o2b5o bo59bo4b4o4bo145bob2o25b2obobobob2o$17b5obo2bob5o59bobo2b4o2bobo146b2o 27b2o2bobob2o$16bob3o8b3obo58b5ob2ob5o174b2o6bo$17b2ob2o6b2ob2o62b2o4b 2o148b3obo24b4o2b5o$16bo3bo8bo3bo59bo2bo4bo2bo147bobo27bo3b5o$16bobo 12bobo59b3o6b3o145b5o25bob3ob3o$251bo27bob2o6bo$16b2ob2o8b2ob2o216bob 2o27b2o$17bo2b2o6b2o2bo217b3o25b2o3bo$16bo2b3o6b3o2bo215b4o26bobo4b3o$ 18b5o4b5o216b3o29b4o3b2o$16b6o6b6o214b4o27bob2o$16b5o8b5o217bo29b3o$ 16bob4o6b4obo217b2o31b3o$18b2obo6bob2o219bo28bob6o$17b2o12b2o215bobo 30b2obob2o$17bo14bo216b4o29bob2o$16b3obo8bob3o214b2ob2o27b2o2b4o$16b3o 12b3o214b2o2bo27bo5b2o$15b6o8b6o212b3ob2o29b3ob2o$15b3obobo6bobob3o 211bob3o30bob3o$16b3ob2o6b2ob3o214b3o32b2ob2o$15b3o5bo2bo5b3o210b2obo 34bob2o$15b3obo2b6o2bob3o212b3o37b2o$17bo2b10o2bo212b3o2bo37bo$19b12o 215b3obo35bob2o$20b10o213bo44b2o$20bob6obo213b2o2bo38b4o$20b10o214b2o 36b6obo$23bo2bo215b3o34b2ob3ob2obo$21b8o212bo37b2ob3obo$21b2ob2ob2o 212bobo36bob3obo$21bobo2bobo214b3o33bob4o$23bo2bo214bobob2o35bo$21b3o 2b3o212b4o34b2o2bo$20b2obo2bob2o211b3ob2o34bo2b2o$19b2obo4bob2o208b2ob 2ob2o35b4o$19bob3o2b3obo208bob3obo35b2ob2o$19b2obo4bob2o210bo38bo2bo$ 21b2o4b2o212b2o40bo$18bo12bo250b2o$19b2o8b2o211b2o$241b3o$241b3o$241b 3o$239bob2o$239b3obo$239b4o$238bo2b2o$238bobo$239b2o$239b2o$241bo$238b o$238b2o$237bo$237bobo$235b5o$234b3o$234bob2o$232b5o$234bo$234bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

In February 2020, AforAmpere found the first knightships in D&N, of speeds (2,1)c/5[2] and (2,1)c/6[3] respectively. On October 26, 2020, Adam P. Goucher found a second, larger (2,1)c/5 knightship using ikpx2,[4] and quickly found multiple tagalongs[5][6] (including a period-10 tagalong[7]) and a wickstretcher[8] based on it. The same day, Goucher found a smaller and thinner (2,1)c/6 spaceship.[9]

On October 27, Goucher found a "growing knightship" - a pattern in which the front end moves at (2,1)c/5 while the back end moves at (2,1)c/6.[10]

References

  1. Peter Naszvadi (August 14, 2020). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
  2. AforAmpere (February 19, 2020). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums
  3. AforAmpere (February 22, 2020). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums
  4. Adam P. Goucher (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  5. Adam P. Goucher (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  6. Adam P. Goucher (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  7. wwei23 (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  8. Adam P. Goucher (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  9. Adam P. Goucher (October 26, 2020). Message in #naturalistic on the Conwaylife Lounge Discord server
  10. Adam P. Goucher (October 27, 2020). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums

External links