OCA:Sqrt replicator rule

B36/S245
x=64, y = 64, rule = B36/S245 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 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 rule Sqrt replicator rule or B36/S245 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. Before the end of 2023, this rule was commonly referred to by the inaccurate name logarithmic replicator rule, due to its namesake object being misidentified.

The cellular automaton has rulestring B36/S245, and differs from the "nearby" Move (B368/S245) in the lack of birth on 8 alive neighbours. 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 and slow 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 ZOOM 10 ]]

The namesake replicator with Θ(√t) growth. It was initially assumed to have log(t) growth, leading to the incorrect and obsolete name. (click above to open LifeViewer) RLE: here Plaintext: here

Its behaviour can be emulated by the following pattern, discovered by AforAmpere on April 11, 2020 in an isotropic non-totalistic rule:

x = 3, y = 2, rule = B2a3eny4at/S01c2ci3i4e5e bo$obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 ]]

A small replicator that emulates the one above (click above to open LifeViewer)

This emulates the 4-state elementary cellular automaton in a range-1 neighbourhood, with rule integer 508169473510261892743356039977344.

Growth bounds

Per the convention in Rule 120, let b₂(n) be the number obtained from writing n in binary and reading this result as a base-4 number. (ie. 11₁₀ = 1011₂, so b₂(7) = 1011₄ = 69₁₀)

In AforAmpere's emulator shown above,

• the bottom edge increases to width 1 on generation 2, then to width n on generation f(n) = 2*b₂(n-1)+b₂(⌊n-12⌋)+5.

(lim infn→∞(f(n)n²) = 34, lim supn→∞(f(n)n²) = 94)

• the top edge increases to width n on generation g(n) = 3*b₂(⌊n2⌋+1)+6*b₂(⌊n+2-2^{⌊log₂(n+2)⌋}2⌋)+2*(n mod 2)+5.

(lim infn→∞(g(n)n²) = 38, lim supn→∞(g(n)n²) = 34)

In the tth iteration, the pattern's total height has the asymptotic bounds 2*√t < h(t) < 2*√5*t√3.

The rule's original name stemmed from this asymptotic growth rate being interpreted as logarithmic in character, rather than square-root growth.

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, c/4 diagonal, and (2,1)c/6.

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 ZOOM 10 ]]

(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 [[ HEIGHT 500 WIDTH 600 THUMBSIZE 2 ZOOM 12 ]]

14c/300o (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: 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 [[ HEIGHT 500 WIDTH 700 THUMBSIZE 2 ZOOM 2 ]]

28c/1200o (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 [[ MAXGRIDSIZE 9 HEIGHT 600 WIDTH 800 THUMBSIZE 2 ZOOM 8 ]]

(click above to open LifeViewer) RLE: here Plaintext: here

On May 28, 2021, Luka Okanishi constructed the following period-408 4c/23 spaceship gun without synthesizing from smaller ships.

x = 110, y = 64, rule = B36/S245 23b2obo2b2obo2bo38bo2bob2o2bob2o$23bob3obobo2b2o38b2o2bobob3obo$13bo2b o6bob3obobo2b2o38b2o2bobob3obo6bo2bo$14b2obo5b2obo2b2obo2bo38bo2bob2o 2bob2o5bob2o$12b3obo76bob3o$15bo78bo12$29bo50bo$28b4o46b4o$27bobo2bo 44bo2bobo$5b3o19bobo2bo44bo2bobo19b3o$3bob2obo19b4o46b4o19bob2obo$5bo 23bo50bo23bo$4b3o47b2o47b3o$52bob2obo$52b2o2b2o$52bob2obo$54b2o$54b2o$ 7b2obo88bob2o$5b4o5bo80bo5b4o$4bo2bo2b2ob2o80b2ob2o2bo2bo$4bo2bo2b2ob 2o80b2ob2o2bo2bo$5b4o5bo80bo5b4o$7b2obo88bob2o2$53b4o$54b2o$52bob2obo$ 53bo2bo$54b2o$35b2o36b2o$35b2o16b4o16b2o$33bo4bo32bo4bo$2b4o28bo2bo34b o2bo28b4o$2bo2bo26b2ob2ob2o13bo16b2ob2ob2o26bo2bo$2bo2bo27bob2obo13bob o16bob2obo27bo2bo$2bo2bo45bobo50bo2bo$3b2o47b2o51b2o$52bobo$52bo5$2bo 2bo14bobo64bobo14bo2bo$2bo2bo14b2obo62bob2o14bo2bo$o2b2o2bo11bo4bo60bo 4bo11bo2b2o2bo$obo2bobo11bo4bo60bo4bo11bobo2bobo$20b2obo62bob2o$2b4o 14bobo12b2o36b2o12bobo14b4o$3b2o29bo2bo34bo2bo29b2o$35b2o36b2o$35b2o 36b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ MAXGRIDSIZE 9 HEIGHT 700 WIDTH 960 THUMBSIZE 2 ZOOM 8 ]]

(click above to open LifeViewer)

On September 23, 2022, FWKnightship constructed a 14c/504 orthogonal replicator-based puffer, which Peter Naszvadi and toroidalet both found spaceship versions of on the following day. Also on September 24, FWKnightship found a second puffer with a speed of 28c/2004 orthogonal.

See also

• Rule 120, another one-dimensional cellular automaton exhibiting Θ(√t) growth.

External links

• Sqrt replicator rule at Adam P. Goucher's Catagolue
• B36/S245 at David Eppstein's Glider Database
• B36/S245 (discussion thread) at the ConwayLife.com forums