Infinite growth

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A finite pattern is said to exhibit infinite growth if its population is unbounded. That is, for any number N there exists a generation n such that the population in generation n is greater than N.

Apparent infinite growth may occur in many cellular automata, and rules in which such patterns are commonplace are known as class three cellular automata. However, in many cases there is no known proof that any such pattern in fact will continue to grow indefinitely (and is not simply a particularly vigorous methuselah). Most patterns known to exhibit infinite growth start growing in a predictable way at some point in their evolution.

In Conway's Game of Life, the first known pattern to exhibit infinite growth was the Gosper glider gun. In 1971, Charles Corderman found that a switch engine could be stabilized by a pre-block in a number of different ways to produce either a block-laying switch engine or a glider-producing switch engine, giving several 11-cell patterns with infinite growth. This record for smallest infinitely-growing pattern stood for more than quarter of a century until Paul Callahan found, in November 1997, two 10-cell patterns with infinite growth (it was later determined that twenty-four 10-cell patterns exhibit infinite growth, with 17 unique pattern types). Nick Gotts and Paul Callahan have since shown that there is no infinite growth pattern with fewer than 10 cells, so the question of the smallest infinite growth pattern in terms of number of cells has been answered completely.

#N 10-cell infinite growth #O Paul Callahan #C A 10-cell infinite growth pattern found in 1997 #C No pattern with fewer cells can exhibit infinite growth x = 8, y = 6, rule = 23/3 6bob$4bob2o$4bobob$4bo3b$2bo5b$obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
Paul Callahan's 10-cell infinite growth pattern. No pattern with fewer cells can exhibit infinite growth
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Growth rates

Infinite-growth patterns can be classified by their growth rate. In any cellular automaton, the maximum rate of growth is determined by the geometry of the space. In cellular automata set in 2D Euclidean space such as Life, the maximum growth rate is quadratic; cubic, quartic, quintic etc. growth require spaces of higher dimensionality.[note 1]

Although the simplest infinite growth patterns grow at a rate that is (asymptotically) linear, many other growth rates are possible. The following table summarizes asymptotic growth rates that have been explicitly constructed in Game of Life:

Classification Growth rate f(t) Examples
Superlinear growth t2 (Quadratic growth) breeder 1, max, mosquito 5, metacatacryst
t3/2 t3/2 infinite growth[2] (see gallery), unnamed[3] (see gallery)
ttlog(t)/2 log(t)/2 infinite growth[2] (see gallery)
tlog(t) tlog(t) growth, Gotts dots
tlog(log(t)) tlog(log(t)) growth
tlog*(n)(t) Sawmill
Linear growth t block-laying switch engine, Gosper glider gun, space rake, blockstacker
Sublinear growth sqrt(t) sqrtgun
t1/3 t1/3 infinite growth[2] (see gallery)
t2/3 t2/3 infinite growth[4]
t3/4 t3/4 infinite growth[4]
t2-n[citation needed] ?
log(t)2 log(t)^2 growth
log(t) (logarithmic) Caber tosser 1
log(n)(t)[citation needed] ?
log*(n)(t) Sawmill
sqrt(log(t)) O(sqrt(log(t)))

It is not difficult to see that quadratic growth is the fastest possible growth rate,[note 2] and many patterns that grow at such speed are now known. Sqrt(Log(T)) is the lowest possible growth rate for any cellular automation. There are patterns that exhibit infinite growth but whose population does not tend toward infinity – see sawtooth. By combining a sawtooth with a pattern that grows infinitely at a different rate, it is possible to construct patterns that grow (for example) logarithmically at some times and linearly at other times. There are even patterns with unknown fate, such as the Fermat prime calculator, for which it is not known if they grow infinitely or not.

Gallery

#N t^1.5 #C Population in generation t is asymptotic to t^(3/2)/sqrt(9720). #O Dean Hickerson, dean.hickerson@yahoo.com (12/6/91) x = 218, y = 170, rule = B3/S23 71b3o11b3o$70bo2bo10bo2bo$40b3o11b3o16bo4b3o6bo$40bo2bo10bo2bo15bo4bo 2bo5bo$40bo6b3o4bo17bo4bo8bo$40bo5bo2bo4bo$41bo8bo4bo2$72bo$71b3o$55bo 14b2obo$54b3o13b3o$54bob2o13b2o9bo$55b3o23b2o$45bo9b2o$45b2o30bo$77bo$ 50bo28bo7b3o$50bo28b2o6bo2bo$38b3o7bo29b2o7bo$37bo2bo6b2o29bo8bo$40bo 7b2o38bo$40bo8bo16b3o3b3o$39bo25bo2bo3bo2bo$53b3o3b3o6bo3bo$52bo2bo3bo 2bo5bo3bo$55bo3bo7bo5bo$55bo3bo20b3o$54bo5bo8b3o8bo$45b3o21bobo9bo$47b o8b3o10b3o$46bo9bobo31bo13bo$56b3o16b3o11b3o11b3o$23bo13bo31b3o3bo12b 2obo5bo4b2obo$22b3o11b3o11b3o16b3o4bo11b3o5b3o3b3o$22bob2o4bo5bob2o12b o3b3o30b2o4bo2b2o3b2o$23b3o3b3o5b3o11bo4b3o36b3o$23b2o3b2o2bo4b2o$30b 3o2$97bo5b3o$69bo26b2o5bo2bo$22b3o5bo37bobo32bo$21bo2bo5b2o26bo9b3o32b o$24bo32bobo8b2o34bo$24bo32b3o$23bo34b2o16b3o$75bo2bo$49b3o26bo$49bo2b o25bo9bo$49bo28bo8b3o$39bo9bo27bo8b2obo$38b3o8bo36b3o$2b3o11b3o19bob2o 8bo36b2o21bo13bo$bo2bo10bo2bo20b3o67b3o11b3o$4bo4b3o6bo20b2o68bob2o4bo 5bob2o$4bo4bo2bo5bo58bo5bo26b3o3b3o5b3o$3bo4bo3bo4bo58b3o3b3o25b2o3b2o 2bo4b2o$7bobo2b2o30bo5bo24b2obo3bob2o31b3o$11bo31b3o3b3o23b3o5b3o$7bob 2o31b2obo3bob2o23b2o5b2o$8b3o6bo24b3o5b3o$8b3o5b3o24b2o5b2o28bo28b3o6b o$9b2o4b2obo60b3o26bo2bo6bo$8bobo4b3o29bo63bo6bo$8bo2bo4b2o28b3o30bobo 29bo3bobo$8bo71bo29bo4bobo$12bo33bobo66b3o$9bo2b2o5bo5bo21bo$9bo2b2o4b 3o3b3o74b3o3b3o$3o7b3o5bob2ob2obo73bo2bo3bo2bo15bo$o2bo6b3o6b3ob3o77bo 3bo17b3o$o18b2o3b2o77bo3bo16b2obo$o101bo5bo15b3o$bo20bo102b2o$21bobo 80b3o$20bo3bo79bobo$22bo81b3o$121bo$7b2o87b3o21b2o$6bobo86bo2bo5b3o13b obo$8bo20b3o66bo6bo$29bo2bo65bo$29bo67bo$29bo167bo$30bo116bo50bo$148bo 45bo3bo$144bo3bo46b4o13b4o$145b4o62bo3bo$137bo2b3o72bo$136b2o2bobo68bo 2bo$137bo2b3o13b2o8bo22bo$145b4o6b4o8bo19bobo19b2o$144bo3bo6b2ob2o3bo 3bo20b2o18b5o$15bo100b3o11b3o15bo8b2o5b4o33bob3o2bo4bo$14b3o98bo2bo10b o2bo14bo51b2o3bo3b3o2bo$14bob2o100bo4b3o6bo51bo14bobo7bo2b2o$15b3o100b o4bo2bo5bo49bobo14bob2o7b2o$15b3o99bo4bo3bo4bo30b2o19b2o15b2o$15b2o 105b4o35b2ob2o35bo$123bo14b2o22bo2bo$137bobo22bo2bo13bo25b2o5b4o$117bo 21bo23b2o12bobo23b2ob2o3bo3bo$116b3o59b2o9b2o5b4o3b4o8bo$115b2obo68b2o b2o3bo3bo4b2o8bo$101bo13b3o25b2o21bo20b4o8bo$100b3o13b2o24bobo22bo20b 2o8bo$99b2obo41bo3bo14bo3bo$99b3o45bo3bo12b4o$99b3o45bo2b2o44bo$100b2o 50bo44bo$109b3o3b3o14b3o13b2o41b2o4bo$108bo2bo3bo2bo13bo2bo55bo4b2o$ 111bo3bo16bo62bo$111bo3bo16bo$110bo5bo16bo9b3o27b3o$144b3o27b3o$112b3o 81b4o$112bobo80bo3bo$112b3o63b4o17bo$177bo3bo16bo$58b3o24b3o11b3o79bo$ 60bo23bo2bo10bo2bo10b3o65bo$59bo27bo4b3o6bo11bo86bo$87bo4bo2bo5bo52b2o 45bo$86bo4bo3bo2bobo49b4ob2o3b4o33bo3bo$91bobob2o30b3o20b6o3b6o33b4o$ 92b2ob2o29bo2bo21b4o4b4ob2o28bo$93b3o33bo33b2o30b2o$86bo42bo64bo$85b3o 41bo68b4o$84b2obo5b2o34bo67bo3bo5bo$84b3o6bo34bo72bo6bo6b2o$85b2o85b3o 25bo3bo3bo4b2ob2o$171b5o29b4o4b4o$171b3ob2o37b2o$73b3o98b2o$75bo132bo$ 74bo132bo3b3o$101b3o82b2o18bo3bob3o$100bo2bo83b2o18bo6b2o$103bo82bo21b 5obo$103bo106b4o$102bo108bo3$215b2o$197b2o14b2ob2o$196b4o13b4o$196b2ob 2o13b2o$198b2o15$109b3o$108bo2bo$111bo$111bo$110bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
t1.5 infinite growth[2]
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RLE: here Plaintext: here
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24b2o8b2o5b3o$364b2o4bo2bo4bo4b2o54b2ob2o84bo14bo9bo2bo48bo30b2o4bo8bo 3b4o4bo2bo39b3o6b2ob2o$370b5o4bo3bo57b2o83b3o15b2o8b2o9b5o5bo60bo9b2o 2bobo3b2o52b2o5b3o$372b2obo3bo2bo5b2o155bo18bo4bo6bo22b2o5bo2bo26b2ob 2o4bo2b2o3bo57b2o9b4o$374bo11b2ob2o178bo4b3o21b4o8bo25b2o8b2o2bobo3b2o 63bo3bo$386b4o132bo24b2o5bo2bo6bo3bo29b2ob2o3bo3bo31bo8bo3b4o66bo$387b 2o134bo22b4o8bo7bo26b2o5b2o5b4o27bo2b2o12b2ob2o61bo2bo$521b3o22b2ob2o 3bo3bo11bo20b2ob2o44b2o15b2o$541b2o5b2o5b4o12bo7b3o5bo3b4o28bo$526bo 12b2ob2o25b3o4bo2b3o3b2obo3b2o27b2o$297b2o28b2o28b2o28b2o14bo13b2o28b 2o28b2o28b2o8bo8bo8bo3b4o33bobo6bo3bo32b2o$297b2o28b2o28b2o28b2o15bo 12b2o7bo20b2o28b2o28b2o9bo11b2o2bobo3b2o34bo2b3o3b2obo3b2o139bo$294bo 107b3o19b2o90b3o2b2o6bo2b2o3bo41b3o5bo3b4o138bobo$292bobo130b2o103b2o 2bobo3b2o49b2ob2o89bo47b2o$293b2o55bo175bo8bo3b4o50b2o91b2o$348bobo 175bo12b2ob2o141b2o$349b2o190b2o$516bo2bo$520bo$516bo3bo$356b2o28b2o 129b4o$297b2o28b2o26bo2bo26bo2bo$296bobo27bobo26bo2bo26bo2bo166bo$296b 2o28b2o28b2o28b2o168bo21bo$554b3o19b2o$577b2o$688bo$688bobo$670bo17b2o $671b2o$670b2o2$298b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o$298bobo 27bobo26bo2bo26bo2bo26bo2bo26bo2bo26bo2bo26bo2bo26bo2bo$299b2o28b2o26b o2bo26bo2bo26bo2bo26bo2bo26bo2bo26bo2bo26bo2bo$358b2o28b2o28b2o28b2o 28b2o28b2o28b2o149b4o$289b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o 28b2o28b2o28b2o28b2o37bo3bo$288bobo27bobo27bobo27bobo27bobo27bobo27bob o27bobo27bobo27bobo27bobo27bobo27bobo41bo$288bo29bo29bo29bo29bo29bo29b o29bo29bo29bo29bo29bo29bo39bo2bo$287b2o28b2o28b2o28b2o28b2o28b2o28b2o 28b2o28b2o28b2o28b2o28b2o28b2o64b2o$351b3o348bo8b2ob2o$353bo346bobo4bo 3b4o$294bo57bo346bo2bo5bo3b2o$294b2o403bo6bo2bo$293bobo3b2o28b2o28b2o 28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o29bo8bo2bo5bo3b2o$ 299b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b2o28b 2o9bobo4bo3b4o$702bo8b2ob2o$713b2o5b2o5b4o$693b3o22b2ob2o3bo3bo$695bo 22b4o8bo$694bo24b2o5bo2bo$391b2o$376bo12b2ob2o323bo$376bo8bo3b4o305b3o 15b2o8b2o$380b2o2bobo3b2o308bo14bo9bo2bo$366b3o2b2o6bo2b2o3bo311bo16b 5o4bo2bo$368bo11b2o2bobo3b2o325b4o3b2ob2o$309bo57bo8bo8bo3b4o327bo4b2o $309b2o25b4o36bo12b2ob2o309b3o$308bobo14bo9bo3bo24bo26b2o5b2o5b4o296bo $326b2o11bo22bo3bo4b3o22b2ob2o3bo3bo295bo$323b3o5b2o2bo2bo28bo5bo22b4o 8bo318b4o$314bo7b3o6b3o28bo4bo4bo24b2o5bo2bo301bo2bo13bo3bo$315bo7b3o 5b2o2bo2bo24b5o345bo16bo$314bo11b2o11bo55bo313bo3bo12bo2bo$325bo9bo3bo 12b2o22b3o15b2o8b2o304b4o$336b4o3b4o4b4o23bo14bo9bo2bo$319bo22bo3bo4b 2ob2o21bo16b5o4bo2bo$312b2o5b2o25bo6b2o40b4o3b2ob2o$309b3ob2o3bobo21bo 2bo52bo4b2o$309b5o67b3o$310b3o70bo$324bo18b2o37bo$324b2o17bo7b2o52b4o$ 323bobo17bo2bo5b2o33bo2bo13bo3bo$344b3o4b2o38bo16bo$330bo15b2o3bo35bo 3bo12bo2bo$330bo57b4o$330bobo$331bo20b2o$331b2o18b4o$351b2ob2o$336b2o 15b2o$334b2ob2o$334b4o$335b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
unnamed t3/2 infinite growth[3]
(click above to open LifeViewer)
RLE: here Plaintext: here
#N t log(t)^2 #C Population in gen t is asymptotic to t log(t)^2 / (12 log(3)^2). #C More specifically, for n>=4, the population in gen 20 3^n + 90 #C is (5n^2 + 43) 3^(n-1) - (15n^4 + 70n^3 + 237n^2 + 254n)/12 + 3440. #C #C In gen 20 3^n - 123 (n>=2), a glider in an exponential aperiodic #C device bounces off a fixed reflector. The bounce turns on a MWSS #C gun and sends a glider toward a diagonal line of boats, as in the #C linear aperiodic pattern. The MWSSs destifle guns produced by a #C stifled breeder. When a glider returns from the nearest boat, the #C MWSS gun is turned off, having produced n+1 MWSSs. In gen t, #C about log(t)/log(3) MWSS pulses have occurred, activating about #C log(t)^2 / (2 log(3)^2) guns, which have produced about #C t log(t)^2 / (60 log(3)^2) gliders. #O Dean Hickerson, dean.hickerson@yahoo.com (4/10/1991) x = 369, y = 240, rule = B3/S23 67b3o11b3o$66bo2bo10bo2bo$69bo4b3o6bo$69bo4bo2bo5bo$68bo4bo3bo2bobo$ 73bobob2o$74b2ob2o$75b3o$68bo$67b3o$66b2obo6bo$66b3o6b2o$67b2o7bo$73bo $71b4obo$72bo2bo$73b3o$60b3o3b3o14b3o$59bo2bo3bo2bo13bo2bo$62bo3bo16bo $62bo3bo16bo$61bo5bo16bo2$37b3o11b3o9b3o$37bo2bo10bo2bo8bobo$37bo6b3o 4bo11b3o$37bo5bo2bo4bo25b3o$38bo4b2obo5bo24bo$63b3o12bo$63b3o2$38bo33b 3o$37b3o32bo5b3o$37bob2o23bo8bo3bo2bo$38b3o22bo16bo$38b2o23b3o14bo$47b 2o31bo$47bo32bo$79bo2$30b3o3b3o5bo8b3o$30bo2bobo2bo5bobo5bo2bo$30bo7bo 5bob2o7bo$30bo7bo5bobo8bo$31bobobobo6b2o8bo2$34bo$33b3o$32b2ob2o$32b2o b2o$33bobo10b2o$34bo11bobo$46bo29b2o$76bo2$41b2o5b3o$18b2o21bobo4bo2bo $17b2ob2o19bo6bo$18b4o3bo22bo$19b2o3bob2o20bo$23bo3bo20bo45bo$19b2o3bo b2o21bo23b2o3b2o14b2o$10bo7b4o3bo47b2o3b2o3b2o6b2o15bo$b2o6bo7b2ob2o 52b5o4bo6b3o15bobo$2ob2o4bo3bo4b2o55bobo13b2o7bo10b2o$b4o4b4o81b2o2b2o 11b2o4b2o$2b2o71b3o16bo4b2o10b2o4bo$33b2o73bobo$32b2o74bo$4bo29bo$3b2o 12b2ob2o24b2o$2b2o14bo2bo23bobo$3b2o13b3o7b2o16bo$13b2o12b2o$13bo15bo 3b2o$32b2ob4o35bo$33b6o34b3o$b2o31b4o34b5o$2ob2o14b2o50b2o3b2o15b2ob2o bo$b4o13b4o70bobobob2o63b4o$2b2o13b2ob2o71bo68bo3bo15bo$18b2o146bo16bo $165bo13bo3bo$103b4o51bo21b4o$76b2o24b6o51bo$61b2o13bo24b8o48b3o$60bob o14b3o20b2o6b2o66b2o$61bo17bo21b8o70bo$102b6o45bo20b2o4b2o$103b4o47bo 18b2o6bo$152b3o17b2o7bo$173bo6bo159bo$341bo$89b2o246bo3bo$85b2o4b4o43b o33b2o8bo155b4o13b4o$85bo3b2ob3o33b3o6bobo31b4o8bo170bo3bo$89bo47b2o 21b4o7b2ob2o3bo3bo174bo$53bobo40b3o60bo3bo9b2o5b4o7b2o54b4o106bo$53bo 3bo96b2o7bo25b2ob2o13b2o37bo3bo15bo$37bo19bo26bo10bo3bo19bo33b4o2bo2bo 26b4o13b4o40bo16bo63bo$35b4o19bo4b2o19b3o8bo3bo20bo31bo3b2o32b2o14b2ob 2o38bo13bo3bo61bobo23bo$36bob2o17bo5bo23bo28bo3bo32b4o2bo2bo10b4o7bo 23b2o24bo29b4o62b2o24bo$34bobob3o7bo4bo3bo28b2o8b3o5b2o11b4o13b4o16b2o 7bo8b6o7b2o48bo120bo$36bob2o9b2o2bobo48bo28bo3bo11b2o8bo3bo8b4ob2o5b2o 43bo5bo119b2o$31b2o2b4o9b2o87bo11bo10b4o12b2o26bo25b6o24b2o92bo$30bobo 4bo58b3o37bo66b4o56bo19bo$30bo123b2o45b5obo50b2o4b2o18bo$29b2o64bo3bo 22b2o30bo45bo6b2o48b2o6bo14bo3bo$86b2o3b2o2bo3bo22bo5b2o3b2o64bo3bob3o 48b2o7bo15b4o13b4o46b2o5b4o$87b5o34bo6bobo64bo3b3o50bo6bo32bo3bo44b2ob 2o3bo3bo$88b3o5b3o26bo2bo6bo65bo99bo44b4o8bo$89bo34b2o7b3o161bo2bo46b 2o8bo$102b2o3b2o16b2o80b2o47b2o8bo$198b4o4b4o45b4o8bo$35b2o65bo5bo82b 2o4bo3bo4b2ob2o36b4o4b2ob2o3bo3bo24b2o3b2o17bo40b2o8bo$36bo152b2ob2o7b o6b2o36bo3bo6b2o5b4o22bo6bobo14bobo39b4o8bo$36bobo6bo23b2o32b2ob2o19b 2o5b4o47bo3b4o7bo42bo6bo20bo17bo2bo6bo15b2o39b2ob2o3bo3bo$37b2o4bobo 23bo35bo13b2o4b2ob2o3bo3bo46bobo3b2o52bo4bo19bobo16b2o7b3o58b2o5b4o$ 41b2o10b2o4bo7bobo48b4o3b4o8bo44b2o3bo55b3o24b2o17b2o$41b2o11b2o2b2o7b 2o49b2ob2o3b2o8bo47bobo3b2o52bo6bo$41b2o10bo7b2o23b3o17bobo11b2o63bo3b 4o50bo8bo95bo$43bobo15b3o42bobo4bobo73b2ob2o52bo5bo93bo2bo13b2o$36bo8b o15b2o7b3o13bobo16bo3bo2bo2b2o74b2o54b6o38b2o5b4o44bo3bo11b2ob2o$37b2o 19b2o12bo12b5o15b5o3bobo167b2o4b2ob2o3bo3bo44bo3bo12bo2bo$36b2o21bo11b o12b2o3b2o13b2o3b2o9b2o160b4o3b4o8bo44bo3bo12bo2bo$84b2o3b2o14b5o8b2ob 2o159b2ob2o3b2o8bo46bobo3b2o9b2o$106b3o9b4o162b2o62bo4bo$107bo11b2o 156bobo$276bo2b2o86bo$277bobo88bo$192b2o28b2o28b2o30b2o49bobo11bo14bo 3bo$192bo29bo29bo29b2ob2o48b2o13bo14b4o$263bo18b4o50bo9bo3bo$86b2o55b 2o119bo18b2o62b4o$86bo56bo116bo3bo$154bobo104b4o78bo$107b2o45bo2bo186b o$55b3o49bo34b2o13b2o6b2o173bo3bo$57bo83bobo11bo3b2o4bo175b4o$56bo74b 2o7bo16b2o173b2o2b3o$44bo86bo8bo2bo10bo2bo174b2o2bobo$43b2o95bo13bobo 35b2o28b2o28b2o28b2o28b2o18b2o2b3o$32b2o8b2o4b2o5bo85bobo47bobo27bobo 27bobo27bobo27bobo27b4o$32bo8b3o4b2o3bobo29bo56b2o47b2o28b2o28b2o28b2o 28b2o27bo3bo$42b2o4b2o2bobo30bobo256bo$43b2o6bo2bo16bo4b2o10b2o4b2o 247bo$44bo7bobo16b2o2b2o11b2o4bo$53bobo12b2o7bo10b2o200b4o$55bo11b3o 15bobo201bo3bo$68b2o15bo86b2o119bo$71b2o99bo119bo$71bo$195bo29bo$193b 2o28b2o$187b2o5bo22b2o5bo22b2o28b2o28b2o$188bo29bo29bo29bo29bo$185b3o 27b3o27b3o27b3o27b3o$185bo29bo29bo29bo29bo19b4o$324bo3bo$328bo$320b2o 2bo2bo$320b3o$320b2o2bo2bo$328bo$324bo3bo6b2o5b4o$325b4o4b2ob2o3bo3bo$ 333b4o8bo$334b2o8bo2$118b4o$117bo3bo219b2o$114bo6bo219bobo$115bo4bo63b o158bo$114b3o68bo155b3o$115bo4bo59bo4bo$114bo6bo59b5o122b6o$117bo3bo 12b2o171bo5bo$118b4o3b4o4b4o176bo$124bo3bo4b2ob2o174bo29b4o$94b2o32bo 6b2o190bo13bo3bo$91b3ob2o30bo200bo16bo$91b5o35bo84bo107bo3bo15bo$92b3o 34bo87bo107b4o$129bo2b2o79bo3bo$128bo5bo27bo51b4o$129bobo2b2o27bo41b2o 2b3o124b4o$129bo3b2o23bo4bo41b2o2bobo123bo3bo$131b3o25b5o41b2o2b3o62b 2o54b2o7bo$214b4o5b2o8bo38b2ob2o52b4o2bo2bo$213bo3bo4b4o8bo33bo3b4o52b o3b2o$134b2o81bo4b2ob2o3bo3bo32bobo3b2o54b4o2bo2bo$133b4o79bo7b2o5b4o 30b2o3bo59b2o7bo$133b2ob2o62b2o65bobo3b2o60bo3bo6b2o5b4o$118b2o15b2o 62bobo66bo3b4o7bo15b2o35b4o4b2ob2o3bo3bo$116b2ob2o80bo70b2ob2o7bo6b2o 7b2o42b4o8bo$116b4o109b2o43b2o4bo3bo4b2ob2o5bo45b2o8bo$117b2o109b2ob2o 48b4o4b4o$229bo2bo57b2o$229bo2bo120bo$201bo28b2o122bo$202bo83b4o64bo$ 196bo5bo83bo2b2o62b2o$197b6o30bo53bo2b2o57bo$234bo52bo2bo28b6o21bo3b3o $230bo3bo53b2o28bo5bo21b2o$214b4o13b4o24b2o63bo$213bo3bo37b4ob2o61bo 29b4o$217bo37b6o77bo13bo3bo$216bo39b4o31b2o21b2o23bo16bo$273b2o14b2ob 2o21b2o18bo3bo15bo$272b4o13b4o21bo21b4o$272b2ob2o13b2o$274b2o72bo$349b o$345bo3bo$346b4o$337b2o2b3o$337b2o2bobo$337b2o2b3o$346b4o$345bo3bo5bo $349bo6bo6b2o$320b3o25bo3bo3bo4b2ob2o$319b5o5b2o22b4o4b4o$319b3ob2o5b 2o30b2o$322b2o5bo$356bo$355bo3b3o$354bo3bob3o$355bo6b2o$356b5obo$358b 4o$359bo3$363b2o$345b2o14b2ob2o$344b4o13b4o$344b2ob2o13b2o$346b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
t log(t)2 infinite growth[2]
(click above to open LifeViewer)
RLE: here Plaintext: here
#C Population is asymptotic to C t^(1/3), where C = (75/16)^(1/3). #C In particular, in gen 720 n^3 + 28620 n^2 + 60n + 3840 #C the population is 15n + 4524. #C This uses two salvos which are sent toward blocks along the #C southeast diagonal. Usually there are 2 blocks on the diagonal. #C At such times, a 3-glider salvo pulls the nearest block 5 units #C and sends back a glider. This is done repeatedly, waiting for the #C return glider from each pull before sending the next one. When the #C nearest block is pulled as far back as possible, it is deleted, and #C a 5-glider salvo is sent toward the other block; the salvo #C duplicates the block, creating a new one 10 units farther away and #C another one 10 units closer. #C Deleting the eater on the western edge of the pattern gives #C population asymptotic to C2 t^(2/3), where C2 = (45/256)^(1/3). #C In particular, in gen 720 n^3 + 28620 n^2 + 60 n + 3840 #C the population is 45 n^2 + 15n + 4522. #C Dean Hickerson, 10/10/2006 x = 493, y = 522, rule = B3/S23 66b2o$67bo2$79bo$77b4o$71b2o5bob2o14b2o$71bo4bobob3o11bo2bo$78bob2o11b o24b2o$77b4o12bo12bo11bobo$79bo5bo7bo11b2o12b3o12bo$84bo9bo2bo11bo10b 3o8b4o$84b3o9b2o11b2o8b3o8b4o$64b5o49bobo9bo2bo10bo$63bob3obo48b2o10b 4o9b2o$64bo3bo62b4o$52b2o11b3o66bo$53bo12bo54bo$121b2o15b2o$139bo$85b 2o$64bo21bo$52bo11bo21bobo9b2o$50b2ob2o8bobo21b2o8bo$62b2ob2o29bo9b2o 29b3o$49bo5bo5bo5bo28bo10bo29b3o$64bo31bo39bo3bo$49b2o3b2o5b2o3b2o29bo $98b2o35b2o3b2o$118b2o$78bo28bo6b4o4b2o$79bo26bo7b3ob2o3bo$77b3o2b2o 22b3o10bo32b2o$61b2o7b2o11b2o68bo$62bo3b2o4b4o6bo$59b3o4bo3b2ob3o$59bo 10bo$108b2o$109bo$47b2o3b2o55bobo10bo$47bo5bo56b2o9b4o$120b2obo5b2o$ 48bo3bo66b3obobo4bo4b3o$49b3o68b2obo10bo3bo12b3o$121b4o8bo5bo10bo3bo$ 122bo10bo5bo9bo5bo$85b2o63bo3bo$74b2o8bobo64b3o$75bo8bo7b2o57b3o$83b2o 7bo2$74bo84b2o$73b2obo58bo24bo$76bo58b2o12b3o8bobo4bo$76bo71b2ob2o3bo 4b2o3bobo$50b2o3b2o16bo2bo49b2o20b2ob2o4b2o5b2o3bo14b2o$74b2o50bo21b5o 3b2o6b2o3bo13bobo$51bo3bo41b3o47b2o3b2o10b2o3bo12b3o$52b3o44bo66bobo 12b3o10bo$52b3o43bo68bo14b3o8b2o$183bobo$184b2o$55bo50bo2bo46b2o$54b3o 45bo2b2o3bo44bo2bo$53bo3bo43b2o7bo44bo$55bo40b2o8b4o45bo$52bo5bo38bo 53bo3bob2o36bo$52bo5bo17bo73b2o5bo37b4o$53bo3bo14bo3b2ob3o114b4o7bo$ 54b3o15b2o4b4o92b2o9bo10bo2bo6bobo$67b2o7b2o79bo12b2obo2bob2o5b2o9b4o 4b2o3bo$68bo27bo17bo41b2o13bo2bo4bo15b4o5b2o3bo10bo$94b2ob2o15b3o58bo 19bo8b2o3bo9b2o$90bo26bo58bo29bobo$88b3o2bo5bo16b2o89bo$87bo109bo$87b 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4bo2bo16bo11b2o102bobo23b2o$120b3o10bobo103bo23bobo$119bo13bobob2o123b o13bobo$99bo19b2o13bobobo114bo8bo2bo10bo2bo$97b2ob2o22b2o3b2o5bo116b2o 7bo16b2o$124bo5bo3bo2bo125bobo11bo3b2o5bo$96bo5bo34bo126b2o13b2o6b2o$ 125bo3bo7bo138bo2bo$96b2o3b2o23b3o8bo105b3o30bobo$117bo16bo2bo$116b3o 17bo$102bo12b5o14bobobo$102b2o10b2o3b2o12bobob2o$101bobo29bobo16b2o$ 134b2o16bo2$116bo$115b2o$115bobo$125bo$124b3o136b2o$101bo21b5o131b4o4b 2o5b3o$100b3o19b2o3b2o130b3ob2o3bo5bo$99b5o8b2o3b2o4b5o136bo10bo9bo$ 98b2o3b2o7b2o3b2o4bo3bo22b2o3b2o127bobo$99b5o9b5o7bo20b2o102b3o19bobo 8bob2o10b2o$99bo3bo10bobo29bo4bo3bo116bo2bo6b2ob2o11bo$101bo50b3o108b 2o10b2o6bob2o$114b3o35b3o108bo9bo3b2o5bobo$275b2o8bo$272bo2bo$102bo48b o120bobo$101b2o22b2o3b2o18b3o$125bobobobo17bo3bo$114bo11b5o20bo$114b2o 11b3o18bo5bo$128bo19bo5bo$149bo3bo$150b3o6$129b3o$128b2ob2o$128b2ob2o$ 128b5o17bo$127b2o3b2o16b2o9$131bo$130b2o5$202bo$202b2o26$200bo$198b3o$ 197bo$197b2o32bo$231b2o5$194b3o$193bo3bo$192bo5bo$192b2obob2o6$194b2o$ 193bo$192b3o16b2o$191bo3bo14bo2bo$190bob3obo16bo$191b5o17bo$210b2obo$ 211bo3$211bo$211b2o$183b2o$184bo$190b2o$191bo$188b3o$188bo4$180b2obob 2o$180bo5bo$181bo3bo$182b3o6$213b2o7b2o54b2o$153b2o59bo7bo55bo$154bo 30bo$154bobo7bobo18bo$155b2o6bo2bo17bobo26b3o26b2o$162b2o10b2o7b2ob2o 26b2o27bo$160b2o3bo9bo6bo5bo22b2o$162b2o21bo25b3o$163bo2bo15b2o3b2o23b obo$164bobo46b2o$2o276b3o$bo223bo51bo3bo$bobo168bo50b3o50bo5bo$2b2o 166b3o49bo53bo5bo$169bo12b2o38b2o15b2o3b2o33bo$169b2o12bo93bo3bo$180b 3o56bo5bo32b3o$180bo98bo$240b2ob2o$242bo$166b3o50b3o58b3o$165b2ob2o48b o3bo38b2o17b3o$165b2ob2o47bo5bo16bo19bo2bo15bo3bo$165b5o47b2obob2o15bo bo18bo$164b2o3b2o67bo3bo17bo17b2o3b2o$238b5o17bob2o$237b2o3b2o18bo$ 238b5o$67b2o170b3o$68bo171bo21bo$68bobo7bo182b2o$69b2o5bobo85b2o$74b2o 12b2o75bo$74b2o13bo72b3o$74b2o86bo56b2o3b2o$76bobo143bo$78bo140bo5bo 57b3o$220b2ob2o57bo3bo$221bobo15bo$222bo16b2o40bo5bo$131bo90bo58b2o3b 2o$131b3o$134bo$133b2o130b2o$265bo$223bo$222b2o$236b2o26b3o$77b2o157bo 27b2o$77bo189b2o$133b2o3b2o13b2o111b3o$93b2o38b2o3b2o13bo111bobo$93bo 40b5o12bobo83bo27b2o12b2o3b2o$78bo12bobo41bobo13b2o83b3o$77b3o11b2o 142bo3bo14bo24bo5bo$76bo3bo54b3o96bob3obo13b3o$75bob3obo46b2o105b5o17b o22b2ob2o179b2o$76b5o47bo127b2o24bo47bo134bo$328bobo$52b2o265bo7bobo$ 53bo264b2o6bo2bo12bo$135bo171b2o8b2o4b2o2bobo11b2o$135b2o146bo23bo8b3o 4b2o3bobo$128b3o125bo5bo19b2o33b2o4b2o5bo101b2o$256b2o3b2o55b2o113bo$ 128bobo188bo113bobo5b2o$127b5o120b3o3b3o173b2o5b3o$126b2o3b2o119bo5b3o 182b2obo11bo$126b2o3b2o120bo5bo71b2o110bo2bo10b2o$331bo111b2obo9b2o8b 2o$27bo111b2o97b3o55b2o34b3o106b3o11b3o9bo$26bobo50b3o57bo97bo3bo55bo 37bo105b2o13b2o$25bo3b2o8bo38bo3bo153bo5bo91b2o121b2o$14bo10bo3b2o5b4o 37bo5bo152b2obob2o38bo176bo$14b2o9bo3b2o4b4o9b2o27b2obob2o172b3o22b3o$ 26bobo6bo2bo10bo103bo102b3o25bo$27bo7b4o112b3o101bo3bo23b2o$36b4o110bo $39bo110b2o102b2o3b2o2$77b2o60b3o146b2o178b2o$78bo59bo3bo97bo47bobo 176bobo$75b3o50b2o3b2o2bo5bo95b2o47bo4b2o3b2o147b2o17bo13bobo$75bo34b 2o18b3o4bo5bo151b3o150bo8bo8bo2bo10bo2bo$111bo17bo3bo160bo3bo158b2o7bo 16b2o$111bobo6b2o8bobo43b2o117bobo169bobo11bo3b2o5bo$112b2o7bo9bo45bo 118bo117b2o52b2o13b2o6b2o$123b2o290bo64bo2bo$123b3o7b2o121bo36b3o40bob o76bobo7b2o30b2o21bobo$123b2o8bo122b2o35b3o39bo2bo77b2o6bobo26b2o2b2o 2b2o$121bo7b2o3b3o38b3o147bo8b2o87b3o4b2o4bo16bobo2bo3bo$79b2o39b2o7bo bo4bo38b3o146bobo5b2o3bo9bo74b3o4bo2b2obobo16b3o$80bo50bo42bo3bo145b2o bo6b2o10b2o75b3o4b3obo3bo16b2o$131b2o6bo5bo27bo5bo111b2o3b2o26b2ob2o6b o2bo85bobo4b2obo3bo$139b2o3b2o28bo3bo113b5o27b2obo8bobo86b2o6b2o3bo$ 175b3o115b3o18b2o8bobo108bobo8b2o$141b3o150bo18bobo9bo9bo100bo9bobo$ 141b3o169bo22bo111bo$142bo169b2o20b3o111b2o3$459b2o$50b2o239b2o80b2o 84bo$50b3o239bo81bo97b2o$52b2obo11bo73bo36b2o109b3o37b3o42bobo8bo85bob o$41bo10bo2bo10b2o73b2o35bo110bo41bo43b2o8bobo67bo14bo10b2o$41b2o9b2ob o9b2o8b2o102b3o148bo55bobo7bo57bobo13bo2bo8bo$50b3o11b3o9bo36b2o66bo 204bo2bo6b2o55bob2o13bo$50b2o13b2o47bo222bo26bo21bobo2b2o4b2o53b2ob2o 14bobo$66b2o46bobo6bo208b3ob2o3bo21b3o19bobo3b2o4b3o53bob2o15b2o$67bo 47b2o4bobo208b4o4b2o20b5o18bo5b2o4b2o50b2o3bobo$120bobo188b2o9b2o12b2o 58b2o7b2o41bobo4bo$119bo2bo188bobo7bobo72bo8bobo40bo$96b2o22bobo179b2o 10bo8bo5bo55bo21bo39b2o$97bo23bobo8b2o168bo8bo2bo13bobo54b2o20b2o$123b o8bobo179bo13b2obo4b2o$134bo176bobo14b2ob2o4bo$134b2o175b2o15b2obo30b 5o$328bobo32b3o$329bo8bo2bob2obo2bo14bo$338b4ob2ob4o$338bo2bob2obo2bo 3$69b2o312bo$67bo2bo90bo217bo3b2ob3o28b2o$66bo13bo79bobo216b2o4b4o28bo $58bo7bo11b4o78b2obo219b2o23b2o5bobo$58b2o6bo12bob2o9b2o19b2o45b2ob2o 4bo237b3o5b2o$67bo2bo6bobob3o9bo13b2o5bo45b2obo4b2o181bo4bo35bo11bob2o 28bo$69b2o8bob2o25bo5bobo6b2o24b2o9bobo186b2ob4ob2o33b2o10bo2bo27b2o$ 78b4o33b2o7bo23bobo10bo189bo4bo26b2o8b2o9bob2o$80bo45b2o20bo234bo9b3o 11b3o9b2o$126b3o18b2o244b2o13b2o10b2o$126b2o264b2o25bo$124bo7b2o258bo$ 123b2o7bobo184b2o$134bo185bo$134b2o$312b2o132bo$79bo23b2o207bo132b2o$ 79b4o20bo197bobo6bobo$80b4o10b2o5bobo11b2o184bo3bo4b2o$69bo10bo2bo9bob o5b2o13bo174b2o12bo$69b2o9b4o8b3o196bo14bo$79b4o8b3o211bo$79bo12b3o 206bo3bo10b2o3b2o$93bobo205bobo13b5o$94b2o221b2ob2o$317b2ob2o$318b3o3$ 89b2o$88bo3bo$87bo5bo7bo219b2o$77bo9bo3bob2o4bobo219bo$77b2o8bo5bo3b2o 12b2o209b3o$88bo3bo4b2o13bo211bo14bo$89b2o6b2o239bo$99bobo119b2o115b3o $101bo119bo$389b2o$390b2o$389bo5$219b2o3b2o$222bo$219bo5bo$220b2ob2o$ 221bobo$222bo$222bo2$191bo$191bobo150bobo$192bobo4b2o18b2o123bo3bo$ 192bo2bo4bo19bo107bo19bo$192bobo22b3o106b4o19bo5bo$180b2o9bobo23bo109b ob2o17bo5b2o$179bobo9bo133bobob3o7bo4bo3bo$179bo147bob2o9b2o2bobo$178b 2o142b2o2b4o9b2o12bo$321bobo4bo25bo$321bo28bo4bo2bo$320b2o28b2obo2bob 2o$354b2o3$359b2o$360b2o$359bo7$340bobo21b2o$340bo3bo19bo$243b2o85b2o 12bo10b2o5bobo$242bobo85bo14bo7bo2bo5b2o$241b3o100bo7bo$232bo7b3o97bo 3bo7bo$232b2o7b3o96bobo9bo$242bobo6b2o100bo2bo$243b2o6bobo101b2o$253bo $253b2o32$235b2o$235bo$232b2o10b2o$231b3o11bo$232b2o$225b2o8bo$224bobo 8b2o$224bo$223b2o14$225b2o$224bobo$214b2o7bo6b2o$214bo8bo2bo2bo2bo$ 223bo6b3o$224bobo5b3o$225b2o6bobo$235bo$235b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
t1/3 infinite growth[2]
(click above to open LifeViewer)
RLE: here Plaintext: here

See also

Notes

  1. Alternatively, a faster-than-quadratic growth (including possibility of exponential growth) may be possible on a two-dimensional hyperbolic plane.[1]
  2. Even if everything expanded at the speed of light, the number of cells that would be "on" would be limited by the number of cells within n generations from the starting region, which is n2.

References

  1. Adam P. Goucher (November 12, 2012). "Self-replication".
  2. 2.0 2.1 2.2 2.3 2.4 2.5 Dean Hickerson's Game of Life Page: Unusual growth rates
  3. 3.0 3.1 Extrementhusiast (July 14, 2013). Re: t^1.5 Growth (discussion thread) at the ConwayLife.com forums
  4. 4.0 4.1 Jormungant (April 15, 2023). Re: Discussion thread for infinite growth patterns (discussion thread) at the ConwayLife.com forums

External links