OCA:Sqrt replicator rule

Logarithmic replicator rule
x=64, y = 64, rule = B36/S245 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 RANDSEED 1729 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]] LifeViewer-generated pseudorandom soup
Rulestring	245/36 B36/S245
Rule integer	26696
Character	Stable
Black/white reversal	B012578/S0134678

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.

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.

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

Spaceships

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.

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

Here is a p23 orthogonal ship gun.

x = 418, y = 425, rule = B36/S245 63bo2bobo2bo$63b4ob4o$65bobobo$64bobobobo108b2obobob2o$63bo2bobo2bo 108b3ob3o$62b3o5b3o108bobobo$62b2o7b2o108bo3bo$64b7o108bo7bo$179b4ob4o $179b2o5b2o$179b9o$181b5o$181b5o$183bo10$65b4o2$64b2o2b2o$64b6o$64bob 2obo$181bo4bo$182b4o133bo2bobo2bo$180b2o4b2o131b4ob4o$319bobobobobo$ 181bo4bo16bo2bobo2bo108bo5bo$182b4o17b4ob4o106b2o7b2o$182b4o19bobobo 108b2o7b2o$204bobobobo108bo2b3o2bo$203bo2bobo2bo107b9o$202b3o5b3o108b 5o$202b2o7b2o$204b7o3$65b4o$65bo2bo$66b2o115b2o$183b2o$102b4o3bo72bo2b o$4b2o76bo19b3o3b2o73b2o$3ob4o15bo56bobobo15b3obob3o33bobobo$bobo3bo 14b3ob2o38bo11b2o3bo16b3obo3b2o31bo3b4o$b2o4b2o12bobob2ob2o35bob2o9b2o 3bo15b4o4bo34b2ob2obo261b4o3bo$2o4b3o11b3obob4o35bob2o9b2o3bo16b3obo3b 2o32b2o4b2o15bob3o152b4o65b4o15b3o3b2o$2bo3b3o11b3obob4o36bo11b2o3bo 16b3obob3o32b2o4bob2o12bo2bobobo12b2o136b6o65bo15b3obob3o$2o4b3o12bobo b2ob2o50bobobo17b3o3b2o33bo2b2ob2o11bo3bobobo11bobo136bo4bo65bo2bo12b 3obo3b2o$b2o4b2o13b3ob2o26b2o26bo19b4o3bo31b2o4bob2o11bo3bobobo11bobo 136b2o2b2o37b2o26bo2bo11b4o4bo$bobo3bo15bo30b2o86b2o4b2o13bo2bobobo12b 2o20b4o113b4o38b2o26bo15b3obo3b2o$3ob4o44b2obo86b2ob2obo16bob3o35bo2bo 113b4o38bob2o23b4o13b3obob3o$4b2o46b4o85bo3b4o55b2o2b2o113b2o39b4o42b 3o3b2o$142bobobo57b2o2b2o200b4o3bo$206b2o$206b2o7$49b4o$48bo4bo$363b2o 4b2o$48b2o2b2o309bo2b2o2bo$49b4o310bo2b2o2bo$50b2o313b4o$365bo2bo$286b 2o$281b2o2b3o$282bobobobo$282b2o4bo17b2o$242b4o3bo31b2ob2o2bo15b4o$97b 3o142b3o3b2o33bo4bo15bo3b2o$96bob2o123b4o13b3obob3o32b2ob2o2bo15bo3b2o $96b3o53b2o70bo15b3obo3b2o32b2o4bo15b4o10b2o$96b2o56bo69bo2bo11b4o4bo 34bobobobo17b2o12bo$153bobo68bo2bo12b3obo3b2o31b2o2b3o31bobo$152bo2bo 53b3o12bo15b3obob3o37b2o30bo2bo$209b2o12b4o15b3o3b2o$209bo32b4o3bo$47b 7o312b3o$45b2o7b2o309b5o$45b3o5b3o307b3o3b3o$46bo2bobo2bo308bo2b3o2bo$ 47bobobobo308b3o2bo2b3o$48bobobo310b2o5b2o$46b4ob4o307bo9bo$46bo2bobo 2bo309b2obob2o$362bob3ob3obo$363bo2bobo2bo13$108bo$108bo$108bo$308b3o 14$49b2obobob2o$50b3ob3o$51bobobo304bo2bobo2bo$51bo3bo304b4ob4o$49bo7b o302bobobobobo$49b4ob4o303bo5bo$49b2o5b2o301b2o7b2o$49b9o301b2o7b2o$ 51b5o304bo2b3o2bo$51b5o304b9o$53bo308b5o13$53b2o$52b4o$52b4o$51bob2obo 306b2o$52b4o306b4o$362bo2bo$361b6o$360b8o$360bob4obo$361b2o2b2o5$8b2o 46b2o$3b2o2b3o44b2ob2o$4bobobobo47bo350bobobo$4b2o4bo16bobo27bo349b4o 3bo$3b2ob2o2bo15b2o2bo26bo304b2o23b3o17bob2ob2o$5bo4bo14b2o3bo330b2ob 2o21bo18b2o4b2o$3b2ob2o2bo14b2o3bo330bo28b2o13b2obo4b2o$4b2o4bo15b2o2b o331bo25b3o14b2ob2o2bo$4bobobobo16bobo332bo25b3o14b2obo4b2o$3b2o2b3o 380b2o14b2o4b2o$8b2o51bo3b4o25bo292bo19bob2ob2o$61b2o3b3o21b2ob3o291b 3o17b4o3bo$62b3obob3o17b2ob2obobo7bo304bobobo$61b2o3bob3o17b4obob3o4b 2obo4b4o$63bo4b4o16b4obob3o4b2obo4bo2bo$61b2o3bob3o17b2ob2obobo7bo6b2o $62b3obob3o19b2ob3o$61b2o3b3o25bo$61bo3b4o270bo19b4o3bo$341bo17b3o3b2o $337b4obo14b3obob3o$315b3o18b7ob2o11b3obo3b2o$111b2o203b2o18b7ob2o10b 4o4bo$110b4o203bo19b4obo14b3obo3b2o$109b2o2b2o226bo15b3obob3o$339bo19b 3o3b2o$109bo4bo244b4o3bo$110b4o9$315b2o$314b4o$313b2o2b2o$312b3o2b3o$ 313bob2obo$314b4o$315b2o5$110b3o$109b5o$107b3o3b3o$107bo2b3o2bo$106b3o 2bo2b3o$107b2o5b2o$106bo9bo$108b2obob2o$106bob3ob3obo198b3o$107bo2bobo 2bo198b5o$312b3o3b3o$312bo2b3o2bo$311b3o2bo2b3o$312b2o5b2o$311bo9bo$ 313b2obob2o$311bob3ob3obo$312bo2bobo2bo24$104b2obobob2o$105b3ob3o$106b obobo$106bo3bo$104bo7bo$104b4ob4o$104b2o5b2o$104b9o$106b5o204b2obobob 2o$106b5o205b3ob3o$108bo208bobobo$317bo3bo$315bo7bo$315b4ob4o$315b2o5b 2o$315b9o$317b5o$317b5o$319bo9$107b4o$107b4o$106bob2obo206b2o$106b6o 206b2o$107bo2bo$106bob2obo205b4o$105b3o2b3o203bob2obo$106bob2obo205b4o $107b4o204bob4obo$108b2o207b4o$63b2o252b4o$58b2o2b3o253b2o$59bobobobo$ 59b2o4bo23b2o$58b2ob2o2bo18b2obo3b4o5b2o6b2o$60bo4bo18bobo4b2o2bo4bobo 5b2o$58b2ob2o2bo18bobo4b2o2bo4bobo4bo2bo$59b2o4bo18b2obo3b4o5b2o6b2o$ 59bobobobo23b2o272b2o$58b2o2b3o298b3o2b2o$63b2o276bobo18bobobobo$317bo 2bo18bob2o19bo4b2o$318bobo18b2obobo17bo2b2ob2o$319bo17b4ob3o17bo4bo$ 317b2o18b4ob3o17bo2b2ob2o$339b2obobo17bo4b2o$339bob2o19bobobobo$341bob o19b3o2b2o$363b2o19$67bo2bobo2bo$67b4ob4o$69bobobo$68bobobobo$67bo2bob o2bo$66b3o5b3o$66b2o7b2o$68b7o17$192bo$159b2o30bobo$85b2o67b2o2b3o15bo bo11bo$83b2ob2o67bobobobo15bo2bo10bo$87bo29b2o36b2o4bo13bo3b2obo$86bo 28b4ob3o31b2ob2o2bo13b2ob3ob2o$86bo11bobo14bo3bobo34bo4bo13b2ob3ob2o$ 93b2ob2ob4o11b2o4b2o32b2ob2o2bo13bo3b2obo$92b3o3bobo2bo10b3o4b2o32b2o 4bo15bo2bo$92b3o3bobo2bo10b3o3bo34bobobobo14bobo$93b2ob2ob4o11b3o4b2o 31b2o2b3o$98bobo13b2o4b2o37b2o$115bo3bobo$115b4ob3o$117b2o12$196b2o$ 195b4o$79b2o114b4o$77b6o$78b4o113bo2bo$77b6o112bo2bo$79b2o113bob2obo$ 195bo2bo$194b6o$77bob2obo112bo2bo$76b8o111bo2bo$76bo2b2o2bo112b2o2$79b 2o$79b2o$79b2o7$194b5o$192b9o$79b3o110bo2b3o2bo$78b5o108b2o7b2o$76b3o 3b3o106b2o7b2o$76bo2b3o2bo108bo5bo$75b3o2bo2b3o106bobobobobo$76b2o5b2o 107b4ob4o$75bo9bo106bo2bobo2bo$77b2obob2o$75bob3ob3obo$76bo2bobo2bo10$ 209b5o$209bo3bo$211bo$210bobo$211bo$209bo3bo$209b5o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]

(click above to open LifeViewer)

External links

Sqrt replicator rule at Adam P. Goucher's Catagolue Sqrt replicator rule at David Eppstein's Glider Database

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