User:Melwin22/Castles/infinite growth

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Even though Castles aren't explosive, it is possible to build patterns that grow infinitely, and even fill the space with 100% density.

Wickstretchers

I know about nine different wickstretchers: five orthogonals - 4c/74, 2c/22, 2c/8, 2c/4, 2c/4, and four diagonals - 2c/14, 2c/18, 2c/18, 2c/48 (some of them in form of their predecessor, including some pentaselah reactions).

x = 148, y = 15, rule = B3678/S135678 23o32bo4bo4bo16bo12bob2o46b2o$o21bo4bo12bo12b2o6b4o16bo13bob2o14bo31b 2o$27bo12bo11b2obo4b2o2b2o10bo20b3o12b2obo28b4o$38bo3bo17b6o9b2obo18b 2o46b2o$40bo9bobob3o2b3o2b3o29b3o14bo31b3o$27bo12bo10bo8bo4bo10bo18b2o b3o42bob2o$49bo3bo39b7o17bo26b3o$25b2ob2o21bo41b7o18bo24b3o$40bo33bo 20bo4bo41b4o$27bo12bo32bo67b3o$134bo5b4o$132b10o$132b9o$134bo2bo2b2o$ 137bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

They can sometimes appear during evolution of random patterns (usually diagonal ones do that). For example, the second diagonal wickstretcher from the image above briefly appears during growth of the monstrosity, in generation 32971, but bumps into another part of the pattern in generation 34360. Fourth diagonal wickstretcher also appears, around generation 37420, but is also destroyed around generation 39230.

Also, some wickstretchers can be made wider or thinner:

x = 32, y = 22, rule = B3678/S135678 3$3b3o2b3o8b3o2b3o$3b3o2b3o8b3o2b3o$4b2o2b2o10b2o2b2o$3b8o9b6o$5b4o9b 10o$5b4o10b8o$5b4o8b12o$6b2o10b10o$4bob2obo6b14o$3b8o5b14o$5b4o6b16o$ 6b2o8b14o$6b2o7bobobobo2bobobobo$5b4o$6b2o$6b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

This 6-cell pattern has the smallest population from all patterns in Castles that grow infinitely. After 1 tick, it has the smallest bounding box (3x5) of all such patterns. It evolves into two orthogonal 2c/22 wickstretchers:

x = 5, y = 3, rule = B3678/S135678 2bo2$2ob2o2$2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Fuses

Fuses can also be constructed. They are the opposite of infinite growth, but fit in here thematically. Eight fuses are shown, burning orthogonally at c, c, 2c/13, 2c/15, 8c/84, 6c/74, 2c/53, and diagonally at 2c/7. All can be stabilized in a way that leaves population of 0 at the end (they are all clean fuses).

x = 144, y = 67, rule = B3678/S135678 $25bo3bo$6bo5bo2b2o3bo3bo2bo2bo4bobobobobobo4b2o7bo2bo2bo6bobo2bobo60b o$7bo3bo3b2o2bo4b7o5b9o5b2o8b5o8b6o59b5o$7bo17b5o5b11o3b5o6b5o7b8o58b 5o$15b2o7b7o5b9o5b4o5b7o7b6o57b5o$7bo7b2o8b5o5b11o3b6o5b5o7b8o56b5o$7b o16b7o5b9o5b4o5b7o7b6o55b5o$15b2o8b5o5b11o3b6o5b5o7b8o54b5o$7bo7b2o7b 7o5b9o5b4o5b7o7b6o53b5o$7bo17b5o5b11o3b6o5b5o7b8o52b5o$15b2o7b7o5b9o5b 4o5b7o7b6o51b5o$7bo7b2o8b5o5b11o3b6o5b5o7b8o50b5o$7bo16b7o5b9o5b4o5b7o 7b6o49b5o$15b2o8b5o5b11o3b6o5b5o7b8o48b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o 47b5o$7bo17b5o5b11o3b6o5b5o7b8o46b5o$15b2o7b7o5b9o5b4o5b7o7b6o45b5o$7b o7b2o8b5o5b11o3b6o5b5o7b8o44b5o$7bo16b7o5b9o5b4o5b7o7b6o43b5o$15b2o8b 5o5b11o3b6o5b5o7b8o42b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o41b5o$7bo17b5o5b 11o3b6o5b5o7b8o40b5o$15b2o7b7o5b9o5b4o5b7o7b6o39b5o$7bo7b2o8b5o5b11o3b 6o5b5o7b8o38b5o$7bo16b7o5b9o5b4o5b7o7b6o37b5o$15b2o8b5o5b11o3b6o5b5o7b 8o36b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o35b5o$7bo17b5o5b11o3b6o5b5o7b8o34b 5o$15b2o7b7o5b9o5b4o5b7o7b6o33b5o$7bo7b2o8b5o5b11o3b6o5b5o7b8o32b5o$7b o16b7o5b9o5b4o5b7o7b6o31b5o$15b2o8b5o5b11o3b6o5b5o7b8o30b5o$7bo7b2o7b 7o5b9o5b4o5b7o7b6o29b5o$7bo17b5o5b11o3b6o5b5o7b8o28b5o$15b2o7b7o5b9o5b 4o5b7o7b6o27b5o$7bo7b2o8b5o5b11o3b6o5b5o7b8o26b5o$7bo16b7o5b9o5b4o5b7o 7b6o25b5o$15b2o8b5o5b11o3b6o5b5o7b8o24b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o 23b5o$7bo17b5o5b11o3b6o5b5o7b8o22b5o$15b2o7b7o5b9o5b4o5b7o7b6o21b5o$7b o7b2o8b5o5b11o3b6o5b5o7b8o20b5o$7bo16b7o5b9o5b4o5b7o7b6o19b5o$15b2o8b 5o5b11o3b6o5b5o7b8o18b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o17b5o$7bo17b5o5b 11o3b6o5b5o7b8o16b5o$15b2o7b7o5b9o5b4o5b7o7b6o15b5o$7bo7b2o8b5o5b11o3b 6o5b5o7b8o14b5o$7bo16b7o5b9o5b4o5b7o7b6o13b5o$15b2o8b5o5b11o3b6o5b5o7b 8o12b5o$7bo7b2o7b7o5b9o5b4o5b7o7b6o11b5o$7bo17b5o5b11o3b6o5b5o7b8o10b 5o$15b2o7b7o5b9o5b4o5b7o7b6o9b5o$7bo7b2o8b5o5b11o3b6o5b5o7b8o8b5o$7bo 17b5o6b9o5b4o6b6o7b6o7b5o$15b2o7b7o4b11o3b5o4b7o8b7o6b5o$7bo7b2o9b3o7b 9o5b4o5b7o7b6o4bob4o$7bo27b11o3b6o6b4o8b6o4b6o$14bo2bo8bobo6bob7obo4b 5o5b6o7b7o4b3o$7bo6b4o8bobo10b3o7b3o10bo11b3o6b3o$6b3o6b2o10bo12bo9b2o 10bo10b2obo6bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Growing spaceships

Combining wickstretchers with fuses results in growing spaceships. First example is the fastest wickstretcher with the fastest non-lightspeed fuse; second example has a frontend and backend with well matched speeds, resulting in a very slow growth rate of 48 cells every 814 generations.

x = 19, y = 13, rule = B3678/S135678 2bo13bo$o3bo$2bo11b2ob2o$bobob3o$16bo$3b2obo$5bo$3bo4$13bo$14bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Failed sawtooth

This is not quite a true sawtooth, but the population graph does show sawtooth-like properties... it's just the population is always growing. In the slow-growing phase, it raises by 14 cells every 12 generations; in the fast-growing phase, it raises by 30 cells every 12 generations. Every next phase is about 3x longer than the previous one.

x = 18, y = 17, rule = B3678/S135678 10bobo$9bo3bo$9b5o$10b3o$10b3o$10b3o$11bo3$9bo3bobo$2obo2b12o$18o$15o$ 2b9o2bobo$8o$5o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Linear spacefiller

A spacefiller that grows linearly is possible in Castles. Smallest predecessors in terms of bounding box (6x6) and population (12) are shown. Notice that the second predecessor uses two pentaselahs.

x = 6, y = 6, rule = B3678/S135678 bob3o$6o$bob3o$3obo$6o$3obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)
x = 8, y = 8, rule = B3678/S135678 bo$2obo2$bo2bo$3bo2bo2$4bob2o$6bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)


Population of this pattern approaches the line p=4t, because four lightspeed signals (marked here as S5) that travel along square borders create on average a cell every turn, and the reaction on the corner adds a constant delay, which becomes more and more insignificant over time. At generation 1E+9, population is about 3.993E+9.

Quadratic growth

No perturbation from behind can affect the 2c/4 wickstretchers; because of that, another reaction can continuously send lightspeed "shoots" at them that fill one quarter of the plane (can be easily mirrored to cover one half of the plane, not sure about a full spacefiller).

19-cell predecessor produces this object after about 700 generations, in a slightly messy way. A cleaner version, which consists of 48 cells and grows quadratically from the very beginning, is shown as well.

x = 6, y = 10, rule = B3678/S135678 bo$4o$b2o$2b2o$b2o$4o$bo2b2o$4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)
x = 11, y = 12, rule = B3678/S135678 bobo$7o$b2ob2o$2bob5o$b3o2b3o$4ob3o$bo4b2o$7b2o$7b3o$6b3o$7b2obo$7b2ob o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)


Shorter side (of the cleaner version) grows with speed of 2c/56, and longer with 2c/4, so the bounding box grows at the rate of t256. It's hard to say how much of this bounding box is covered, but it's definitely more than 12, so population grows faster than t2112. At generation 1M, population is about 1.286E+10, so it looks like the actual growth is about t277.