OCA:sqrt replicator rule

B36/S245
x=0, y = 0, 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 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, 2c/7 orthogonal, c/3 diagonal, c/4 diagonal, c/5 diagonal, c/6 diagonal, (2,1)c/5 and (2,1)c/6.

type	speed	period	first known	discoverer	minimal known	discoverer
elementary	c/2	2			17P2H1V0	unknown
		4	51P4H2V0	unknown	26P4H2V0	DroneBetter, 2024
		6			48P6H3V0	DroneBetter, 2024
		8			43P8H4V0	DroneBetter, 2024
		10			156P10H5V0	DroneBetter, 2024
		12			73P12H6V0	DroneBetter, 2024
		20			43P20H10V0	Jason Summers, 2002
		52			86P52H26V0	Jason Summers, 2002
	c/3	3			28P3H1V0	unknown
		6	71P6H2V0	DroneBetter, 2024	61P6H2V0	DroneBetter, 2024
		9			154P9H3V0	DroneBetter, 2024
	c/4	4			13P4H1V0	unknown
	c/5	5	246P5H1V0	unknown	69P5H1V0	JMM, 2024
	2c/5	5	380P5H2V0	unknown	208P5H2V0	Darren Li, 2024
	c/6	6			32P6H1V0	unknown
	c/7	7			44P7H1V0	Evan Clark, 2006
	2c/7	7			504P7H2V0	DroneBetter, 2024
		14			68P14H4V0	DroneBetter, 2024
	4c/23	23			14P23H4V0	unknown
	c/3d	3			60P3H1V1	unknown
	c/4d	4			60P4H1V1	unknown
	c/5d	5			146P5H1V1	DroneBetter, 2024
	c/6d	6			718P6H1V1	DroneBetter, 2024
	c/7d	7			Jellyfish (5P7H1V1)	Nathan Thompson, 1994
	(2,1)c/5	5			544P5H2V1	DroneBetter, 2024
	(2,1)c/6	6			421P6H2V1	LaundryPizza03, 2020
corderoid	7c/150	300	58P300H14V0	Dean Hickerson, 1997	52P300H14V0	Benet Nasch, 2022
	7c/252	504			2715P504H14V0	FWKnightship, Peter Naszvadi, toroidalet, 2022
	7c/501	2004			9980P2004H28V0	FWKnightship, Peter Naszvadi, 2022
	7c/600	1200			280P1200H28V0	Dean Hickerson, 1997

x = 761, y = 182, rule = B36/S245 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#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 WIDTH 3062.5 HEIGHT 742 ZOOM 4 ]]

known elementary spaceships of minimal width and population for each speed, period and symmetry (click above to open LifeViewer)

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, which Naszvadi completed into a spaceship two days later.

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