OCA:Logarithmic replicator rule

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Logarithmic replicator rule
Rulestring 245/36
Rule integer 26696
Character Stable
Black/white reversal B012578/S0134678
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The logarithmic replicator rule is a Life-like cellular automaton in which cells survive from one generation to the next if they have 2, 4, or 5 neighbours and are born if they have 3 or 6 neighbours. It is extremely similar to Move, differing only by the B8 transition. The time required to stabilize is generally much shorter than in Conway's Game of Life.

On August 19, 2020, Peter Naszvadi constructed a Rule 110 unit cell in B36/S245, proving the rule Turing-complete.[1]

Notable patterns

The replicator

The name of this rule comes from an elementary replicator first discovered by Mark Niemiec. Unlike other replicators, (such as the one from HighLife) this one does not reproduce itself cleanly, instead leaving oscillators behind which result in a more chaotic growth pattern.[2]

x = 19, y = 3, rule = B36/S245 6o7b6o$o4bo7bo4bo$b4o9b4o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
The namesake logarithmic replicator. (click above to open LifeViewer)
RLE: here Plaintext: here


The rule has several known elementary spaceships, the smallest ones having speeds of c/4 orthogonal, 4c/23 orthogonal, and c/7 diagonal, shown below. Other known elementary spaceship speeds include c/2 orthogonal, c/3 orthogonal, c/5 orthogonal, 2c/5 orthogonal, c/6 orthogonal, c/7 orthogonal, c/3 diagonal, and c/4 diagonal.[3]

x = 7, y = 27, rule = B36/S245 b2o3bo$3ob3o$2bob3o$6bo7$3o$b2o$5bo$4b2o$4b2o$5bo$b2o$3o7$3o$2o$o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

In 1997, Dean Hickerson discovered two replicator-based spaceships traveling at 7c/150 orthogonal and 7c/300 orthogonal respectively:

x = 38, y = 17, rule = B36/S245 15bobo$16bo4bobo$12bo3bo4bobo12bo$12bo2bobo2bo3bo9b2obo$12bo2bobo2bo3b o9b2obo$12bo3bo4bobo12bo$16bo4bobo$15bobo4$b4o$3bo17bo$o2bo15b2obo$o2b o15b2obo$3bo17bo$b4o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 280, y = 163, rule = B36/S245 170bo$171bo$170bo16bo$127b3o38b2obo14bob2o$168b2obo14bob2o$170bo16bo$ 171bo$170bo7$171bobo$170b2obo$169bo2bo15b2o$169b3o16bobo$169b3o16bobo$ 169bo2bo15b2o$170b2obo$171bobo6$236bobo$170bo32bo15b2o14bo2b2o36b2o$ 171bo31bo14bobo14bo3b2o34bobo$170bo16bo15bo14bobo14bo3b2o34bobo$168b2o bo14bob2o29b2o14bo2b2o36b2o$127b3o38b2obo14bob2o46bobo$170bo16bo$171bo $170bo6$237bo$94bo27bo27bo27bo41bo15bob2o37bo$93bob2o24bob2o24bob2o24b ob2o38bob2o13b5o35bob2o$93bob2o24bob2o24bob2o24bob2o38bob2o13b5o35bob 2o$94bo27bo27bo27bo41bo15bob2o37bo$237bo4$4o24b4o24b4o48bo3b2o$81b2o 25bob2o2b2o$10b2o26b2o26b2o13b3o23b3o6b2o19b2o$9bobo25bobo25bobo38bobo b2ob3o2b2o16bobo$9bobo25bobo25bobo38bobob2ob3o2b2o16bobo$10b2o26b2o26b 2o39b3o6b2o19b2o97bobo$108bob2o2b2o103b2o14bo2b2o36b2o$108bo3b2o89bo 14bobo14bo3b2o34bobo$203bo14bobo14bo3b2o34bobo$203bo15b2o14bo2b2o36b2o $236bobo2$110bo$108b2obobo$20b2o86bo3b3o$19b4o84b2obo2bobobo$19b2o84b 2obo3bo2b4o18bo$20bo83bobo5bob4obo15b2obo$104bobo5bob4obo15b2obo$105b 2obo3bo2b4o18bo$85bo21b2obo2bobobo$85bo22bo3b3o$85bo22b2obobo$110bo15$ 26bo$25bo59bo$26bobo56bo18bo$27bo57bo18bo$105b2o16bo13bo$102bob4o13b2o bo10b2obo$102bob4o13b2obo10b2obo$105b2o16bo13bo$104bo$104bo4$241bo$ 203bo15b2o12b3ob4ob2ob2o29b2o$203bo14bobo12bo3bo7bobo27bobo$203bo14bob o12bo3bo7bobo27bobo$106bo11b2o99b2o12b3ob4ob2ob2o29b2o$10b2o26b2o26b2o 37b3o15b2o12b2o102bo$9bobo25bobo25bobo36b2o12b2o2bobo11bobo$9bobo25bob o25bobo36b2o12b2o2bobo11bobo$10b2o26b2o26b2o13b3o21b3o15b2o12b2o$81b2o 23bo11b2o$4o24b4o24b4o3$234bo$235bo11bo$94bo27bo27bo27bo41bo14bo3bo4bo bo30bo$93bob2o24bob2o24bob2o24bob2o38bob2o11b2o4b2o2bo3bo27bob2o$93bob 2o24bob2o24bob2o24bob2o38bob2o11b2o4b2o2bo3bo27bob2o$94bo27bo27bo27bo 41bo14bo3bo4bobo30bo$235bo11bo$234bo4$170bo2$163b2o6bob3o$161b2o7bo3bo 12bo$127b3o30bo3bo2b6obob2o8bob2o51bo$160bo3bo2b6obob2o8bob2o29b2o12b 3ob4ob2ob2o29b2o$161b2o7bo3bo12bo15bo14bobo12bo3bo7bobo27bobo$163b2o6b ob3o27bo14bobo12bo3bo7bobo27bobo$203bo15b2o12b3ob4ob2ob2o29b2o$170bo 70bo5$169bo3b2o$169b3ob2obo$164bob2o4b3obo$164bo3b2o2b2ob3o10b2o$164b 3o4bobo14bobo$164b3o4bobo14bobo$164bo3b2o2b2ob3o10b2o$164bob2o4b3obo$ 169b3ob2obo$169bo3b2o5$170bo2$163b2o6bob3o$161b2o7bo3bo12bo$160bo3bo2b 6obob2o8bob2o$127b3o30bo3bo2b6obob2o8bob2o$161b2o7bo3bo12bo$163b2o6bob 3o2$170bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Linear growth

Replicators can also be used to create a gun for the c/7 diagonal ship:

x = 74, y = 45, rule = B36/S245 55b2o$55bobo$16bo38bobo$14b4o37bobo$13bo2bobo36bobo$13bo2bobo36b2o$14b 4o$16bo9$69b4o$68bo4bo$2b2o64b6o$bo2bo$bo2bo$6o$bo2bo28b6o$2b2o29bo4bo $34b4o11$50b2o$49bobo$49bobo$49bobo$23bo25bobo$22b4o24b2o$21bobo2bo$ 21bobo2bo$22b4o$23bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here


  1. Peter Naszvadi (August 19, 2020). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
  2. David Eppstein. "Replicators: B36/S245". Replicators. Retrieved on June 2, 2019.
  3. David Eppstein. "B36/S245". Retrieved on June 2, 2019.

External links

  • B36/S245 (discussion thread) at the ConwayLife.com forums