For example, this approach doesn't distinguish between a honeyfarm (four beehives) and two traffic lights (eight blinkers), because each of them has constant population of 24 alive cells.
In general, which population sequences are possible in CGoL?
Which population sequences are possible for specific types of CGoL evolutionary sequences?
I think the following gives all possible population sequences where the first term is less than 4:
Code: Select all
0, 0, 0, 0, ... (empty space)
1, 0, 0, 0, ... (dot / monomino)
2, 0, 0, 0, ... (domino, duoplet, two isolated dots)
3, 0, 0, 0, ... (obobo!)
3, 1, 0, 0, ... (obo2$o! ; o$bo$2bo!)
3, 2, 0, 0, ... (obo$o! ; obo$bo!)
3, 3, 3, 3, ... (blinker)
3, 4, 4, 4, ... (L-triomino)
There is a known 44-bit "grow-by-one" pattern, and there is a known 53-bit "grow-by-one" object. Quoting from the source page:
Constant-population sequencesGame of Life News: Grow-By-One Pattern wrote: In his continuing quest for Diagonal Spaceships, Nicolay Beluchenko has also found what is currently the smallest known "Grow-By-One" pattern. This type of pattern is one whose population growth rate is exactly linear, without any fluctuations, adding a single bit every generation. Shown here is a slight improvement by David Bell, which starts a generation earlier than Beluchenko's orginal pattern, with a population of 44 at generation 0. (The purpose of the Lightweight Spaceship is to smooth out the fluctuations in the paired wickstretcher's population.)
The second pattern shown here is a version which is also a single object (starting with a population of 53).
(Posted by H Koenig on 2005 September 10)
Code: Select all
#C 2005-09-10-GrowByOne.rle #C by N.Beluchenko/D.Bell x=17, y=15, rule=B3/S23 8b2o$7b2o$9bo$11b2o$10bo2$9bo2b2o$b2o5b2o4bo$2o5bo5bo$2bo4bobo3b2o$4bo2bo4b 2obo$4b2o7b2o$8bo4bob2o$7bobo2bob2o$8bo! x=35, y=21, h=-6, v=-6, rule=B3/S23, gen=24 8b2o$7b2o$9bo$11b2o$10bo2$9bo2bo$b2o5b2o2bobo$2o5bo5bobo$2bo4bobo4bobo$4b 2o2bo6bobo$4b2o10bobo$8bo8bobo$7bobo8bobo$8bobo8bo$9bobo$10bobo18b2o$11bob o16b4o$12bobo15b2ob2o$13bobo16b2o$14bo! CB 1,1,1,4 L CM x=0 y=17 text="Gen 0" L CM x=22 y=17 text="Gen 24" color=(0,192, 0)Code: Select all
#C 2005-09-10-GrowByOneObj.rle #C by N.Beluchenko/H.Koenig x=15, y=22 7b2o$7bobo$7bo$10bo$10b2o$10b2o$9b2o$2o6b2o3bo$obo3b2o2bobobo$o5b3o4bo$3b o$3b2o$7bo$6bobo$7bo$4bo$4b6o$10bo$6bo3bo$6b2obo$10b3o$12bo! x=3, y=2, h=7, v=17, color=(0,255,0) bo$obo! CB 1,1,1,1
The blinker is a constant-population oscillator, with population 3 (even though the smallest still-life patterns, the block and the tub, are 4-bitters).
The glider is a constant-population spaceship (5 alive cells in every phase).
What are other small constant-population objects (not counting the still-life objects)?
Are there any constant-population collisions involving only spaceships, oscillators and/or still-life objects?
Are there any constant-population glider collisions?
The following well-known reaction unfortunately fails to be an example:
Code: Select all
x = 5, y = 10, rule = B3/S23
bo$2bo$3o4$3b2o$4bo$b3o$bo!
#C [[ GRID SHOWGENSTATS ]]
confocaloid wrote: ↑February 18th, 2025, 2:40 pmI think one needs to clarify that the still life must be a strict object. Otherwise two well-separated solutions would in many cases count as another solution.get_Snacked wrote: ↑February 18th, 2025, 1:59 pmthe tub is the only synthesisable still life (S) where all predecessors of S that are not S itself have a greater population than S.
EDIT: the snake is a counterexample. other than that, are there any other still lifes that satisfy this condition?
[...]
hotdogPi wrote: ↑February 18th, 2025, 3:24 pmWe already know that a clock predecessor must contain 8 cells or more or a clock (7 or less: itself alone or itself plus a dot anywhere in the universe).confocaloid wrote: ↑February 18th, 2025, 2:40 pmSince the requirement is stated in terms of population, it might be interesting to relax the idea so that it is about constant-population strict objects (including but not limited to strict still-life objects). The blinker would count as an additional counterexample (constant population 3, every predecessor that isn't a blinker would have at least 4 alive cells) unless I'm missing something. The glider wouldn't count, it has a constant population 5, but there is a 5-bit predecessor that isn't a glider:Code: Select all
x = 3, y = 3, rule = B3/S23 o$2bo$3o!
Other related discussionconfocaloid wrote: ↑January 13th, 2025, 9:08 amWhat are lowest-constant-population "completions" of p5 orthogonal c/5 spaceships by collections of p5 oscillators?
Here are some (likely suboptimal) solutions:
Spider is "completed" into a constant-population pattern (160 alive cells in every generation):Kermit is "completed" into a constant-population pattern (177 alive cells in every generation):Code: Select all
x = 63, y = 28, rule = B3/S23 11bo35bo$9b3o35b3o$8bo41bo$8b2o39b2o$7b2o$5b3o44bobo$3b2o2b2o42b4o$4b 3o44bobo$3b2o$2b2o52bobo$3bo51b4o$3o52bobo$o$59b2o$59bo$60b3o$62bo4$4b 2o4b2o13bo7bo13b2o4b2o$4bobo2bobo7b2obobob2o3b2obobob2o7bobo2bobo$6bo 2bo6b3obob3o9b3obob3o6bo2bo$5bo4bo5bo3bobo5bobo5bobo3bo5b2o2b2o$5b6o9b 2o6bobo6b2o8b3o2b3o$6b4o7b2o9bobo9b2o7bo2bo$17b2ob2o15b2ob2o$21bo15bo! #C [[ SHOWGENSTATS ]]Tarantula is "completed" into a constant-population pattern (181 alive cells in every generation):Code: Select all
x = 75, y = 21, rule = B3/S23 bo19bo$2o19b2o$o2bo4b2o3b2o4bo2bo$4bob2obo3bob2obo$3bo15bo9b2o4b2o2b2o 8bo9bo$5b4o5b4o11bobo2bobo2bo2bo6b3o7b3o$9b2ob2o17b4o5bobo9bo9bo$29bo 2b2o2bo14b2o8b2o$10bobo16bo2b2o2bo5bobo17b2o$9bo3bo17bo2bo18b2obo6b3o$ 5b5o3b5o26bobo6bo2bo5b2o2b2o$4b2o2b2o3b2o2b2o34bob2o7b3o$4bo5bobo5bo 27bob2o16b2o$2b4o2bobobobo2b4o36b2o8b3o$9bo3bo34bobo6bo8b2o2b2o$49b2o 7b3o7b3o$60bo9b2o$71b2o$71bo$72b3o$74bo! #C [[ SHOWGENSTATS ]]Code: Select all
x = 60, y = 30, rule = B3/S23 2o4b2o$obo2bobo$2bo2bo$bo4bo$b6o7b2o10bo$2b4o9bo8b3o$12b2o9bo$24bo$10b obo11bo$19b2o$8bobo7bo4bo$21b2o$6bobo8bo$6bo10bo10bo$4bo13bo7b3o$4b2o 9b3o7bo$13b3o9b2o$12bo11b2o$12b2o8b3o$11b2o7b2o2b2o$9b3o9b3o15bo15bo$ 7b2o2b2o7b2o16b3o13b3o$8b3o7b3o23b3ob3o$7b2o7b2o2b2o14bo6bobo3bobo6bo$ 6b2o9b3o15bobobo3bo7bo3bobobo$7bo8b2o20bob2o2b3ob3o2b2obo$4b3o8b2o18bo 23bo$4bo11bo18b2o4b2o9b2o4b2o$13b3o19b2o4b2o9b2o4b2o$13bo! #C [[ SHOWGENSTATS ]]
confocaloid wrote: ↑July 11th, 2024, 8:18 amThis p3 oscillator contains two independent instances of the rotor "cuphook", and has (lexicographically first) population signature (20, 21, 22):hotcrystal0 wrote: ↑July 11th, 2024, 7:53 amThis 2c/4 spaceship starts at 41 cells, and in the following phases grows by one cell per tick until it restarts at 41:Are there any similar oscillators/spaceships, growing by 1 cell every generation until it restarts in its cycle?Code: Select all
x = 17, y = 11, rule = B3/S23 4b4o$4bo3bo$4bo$2b3o3bo$b6o$2ob2o$b2o3b2o$5b3o3b5o$5b3o2bo5bo$10bo$10b o5bo!Two other examples are 1-2-3 and 1-2-3-4.Code: Select all
x = 9, y = 7, rule = B3/S23 4b2o$2o3bo$o$b7o$8bo$3bobob2o$3b2o!
Can anyone find or engineer a p5 oscillator with population signature (k, k+1, k+2, k+3, k+4)?
edit 1: here is an engineered p5 example (5blink + 2 hearts + 2 fumaroles), population signature (125, 126, 127, 128, 129):Code: Select all
x = 42, y = 25, rule = B3/S23 34bo$18bo14bobo$17bobo14bo$18bo$32b5o$2o2b3o9b5o10bo5bo$obobo11bo4bo 12bo2bo2bo$bo14b3o2bo2bo7b2obobobobo$2bobo12bobobobobo8bo2bo2bo$7bo10b 2obo2bo9bo2bo$2bob3obo10b3o14bo$bo2b2o2bo$2b2obo$4bo$5bobo$2bo$bobo$2b o2$17b2o4b2o4b2o4b2o$17bobo2bobo4bobo2bobo$19bo2bo8bo2bo$18bo4bo6bo4bo $18b6o6b2o2b2o$19b4o9b2o!
forum post 196451 wrote: ↑October 23rd, 2024, 5:30 pmFrom: Allan Wechsler
Date: Fri, 10 Apr 1992 11:01-0400
Subject: Missions
I think this is the first time that so many influential Life hackers
have had such fast access to a common forum. Wouldn't it be a good
occasion to restate some of the most outstanding problems? Here are a
few that spring to mind:
[...]
5. Characterize the functions F(t) such that Life patterns exist whose
populations grow as O(F(t)). Provide explicit examples where possible.
[...]
