User:Melwin22/Workspace: Difference between revisions

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yhhhhh this looks very weird but it fits on the page...
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Any pattern that doesn't stabilize eventually turns into an oscillator. For large regions this can take a while, but because the space inside any wall is finite, it has finite configurations and must eventually become periodic.
Any pattern that doesn't stabilize eventually turns into an oscillator. For large regions this can take a while, but because the space inside any wall is finite, it has finite configurations and must eventually become periodic.


Walled Cities allows for the existence of oscillators with the smallest possible [[rotor]] (and lowest possible [[heat]]), which can be attached to different stators:
Walled Cities allows for the existence of oscillators with the smallest possible [[rotor]] (and lowest possible [[heat]]), which can be attached to different [[stator]]s:


{{EmbedViewer
{{EmbedViewer
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This happens for majority of [[apgsearch]]-size soups, so simple objects in this rule are very difficult to find using apgsearch.
This happens for majority of [[apgsearch]]-size soups, so simple objects in this rule are very difficult to find using apgsearch.


Periods of oscillators emerging from hollow rectangles are an extensively researched topic in Walled Cities.<ref>{{LinkForumThread|p=226491|format = ref|title=Re: Miscellaneous Discoveries in Other Cellular Automata}}</ref> If a rectangle has sides {{times|m|n}}, it will produce an oscillator with period listed in the table:
Periods of oscillators emerging from hollow rectangles are an extensively researched topic in Walled Cities.<ref>{{LinkForumThread|p=226491|format = ref|title=Re: Miscellaneous Discoveries in Other Cellular Automata}}</ref> A {{times|3|n}} rectangle always results in a p2 oscillator, with every cell inside flashing on in one phase and flashing off in the other phase. Otherwise, if a rectangle has sides {{times|m|n}}, it will produce an oscillator with period listed in the table:


{|class = "wikitable"
{|class = "wikitable"
|-
|-
! !!colspan="22"|m
! !!colspan="21"|m
|-
|-
!n ||3 ||4 ||5 ||6 ||7 ||8 ||9 ||10 ||11 ||12 ||13 ||14 ||15 ||16 ||17 ||18 ||19 ||20 ||21 ||22 ||23
!n ||4 ||5 ||6 ||7 ||8 ||9 ||10 ||11 ||12 ||13 ||14 ||15 ||16 ||17 ||18 ||19 ||20 ||21 ||22 ||23
|-
|-
|'''3''' ||2
|'''4''' ||2
|-
|-
|'''4''' ||2 ||2
|'''5''' ||4 ||3
|-
|-
|'''5''' ||2 ||4 ||3
|'''6''' ||2 ||2 ||5
|-
|-
|'''6''' ||2|| 2|| 2|| 5
|'''7''' ||4 ||9 ||4 ||7
|-
|-
|'''7''' ||2|| 4|| 9|| 4|| 7
|'''8''' ||4 ||7 ||14 ||7 ||7
|-
|-
|'''8''' ||2|| 4|| 7|| 14|| 7|| 7
|'''9''' ||4 ||5 ||5 ||11 ||17 ||10
|-
|-
|'''9''' ||2|| 4|| 5|| 5|| 11|| 17|| 10
|'''10''' ||6 ||8 ||14 ||28 ||11 ||10 ||17
|-
|-
|'''10''' ||2|| 6|| 8|| 14|| 28|| 11|| 10|| 17
|'''11''' ||4 ||10 ||7 ||2 ||10 ||20 ||56 ||17
|-
|-
|'''11''' ||2|| 4|| 10|| 7|| 2|| 10|| 20|| 56|| 17
|'''12''' ||4 ||11 ||19 ||40 ||50 ||41 ||24 ||3 ||48
|-
|-
|'''12''' ||2|| 4|| 11|| 19|| 40|| 50|| 41|| 24|| 3|| 48
|'''13''' ||4 ||6 ||21 ||32 ||11 ||58 ||222 ||69 ||716 ||56
|-
|-
|'''13''' ||2|| 4|| 6|| 21|| 32|| 11|| 58|| 222|| 69|| 716|| 56
|'''14''' ||10 ||26 ||18 ||51 ||18 ||118 ||384 ||31 ||176 ||504 ||31
|-
|- style="writing-mode: vertical-rl"
|'''14''' ||2|| 10|| 26|| 18|| 51|| 18|| 118|| 384|| 31|| 176|| 504|| 31
|'''15''' ||4 ||22 ||46 ||2 ||48 ||169 ||219 ||748 ||2004 ||1041 ||453 ||58
|-
|- style="writing-mode: vertical-rl"
|'''15''' ||2|| 4|| 22|| 46|| 2|| 48|| 169|| 219|| 748|| 2004|| 1041|| 453|| 58
|'''16''' ||4 ||66 ||46 ||54 ||69 ||176 ||85 ||461 ||2073 ||2412 ||2979 ||9811 ||57
|-
|- style="writing-mode: vertical-rl"
|'''16''' ||2|| 4|| 66|| 46|| 54|| 69|| 176|| 85|| 461|| 2073|| 2412|| 2979|| 9811|| 57
|'''17''' ||4 ||54 ||32 ||185 ||88 ||46 ||468 ||262 ||220 ||4746 ||1612 ||11433 ||62146 ||518
|-
|- style="writing-mode: vertical-rl"
|'''17''' ||2|| 4|| 54|| 32|| 185|| 88|| 46|| 468|| 262|| 220|| 4746|| 1612|| 11433|| 62146|| 518
|'''18''' ||14 ||64 ||28 ||4 ||199 ||47 ||373 ||164 ||2241 ||3158 ||24243 ||521 ||63817 ||54080 ||677
|-
|- style="writing-mode: vertical-rl"
|'''18''' ||2|| 14|| 64|| 28|| 4|| 199|| 47|| 373|| 164|| 2241|| 3158|| 24243|| 521|| 63817|| 54080|| 677
|'''19''' ||4 ||85 ||18 ||79 ||33 ||353 ||101 ||3127 ||2106 ||10938 ||4270 ||57784 ||117911 ||52951 ||385624 ||674
|-
|- style="writing-mode: vertical-rl"
|'''19''' ||2|| 4|| 85|| 18|| 79|| 33|| 353|| 101|| 3127|| 2106|| 10938|| 4270|| 57784|| 117911|| 52951|| 385624|| 674
|'''20''' ||4 ||20 ||18 ||208 ||404 ||512 ||339 ||6784 ||14769 ||13819 ||2388 ||46276 ||218102 ||594883 ||346159 ||132674 ||5160
|-
|- style="writing-mode: vertical-rl"
|'''20''' ||2|| 4|| 20|| 18|| 208|| 404|| 512|| 339|| 6784|| 14769|| 13819|| 2388|| 46276|| 218102|| 594883|| 346159|| 132674|| 5160
|'''21''' ||4 ||77 ||19 ||820 ||486 ||696 ||1518 ||6519 ||5875 ||8649 ||105386 ||41636 ||219190 ||380124 ||1601041 ||4675530 ||1513062 ||22118
|-
|- style="writing-mode: vertical-rl"
|'''21''' ||2|| 4|| 77|| 19|| 820|| 486|| 696|| 1518|| 6519|| 5875|| 8649|| 105386|| 41636|| 219190|| 380124|| 1601041|| 4675530|| 1513062|| 22118
|'''22''' ||18 ||71 ||41 ||1398 ||204 ||2611 ||4128 ||17329 ||14684 ||12556 ||177336 ||22508 ||446333 ||302811 ||2885033 ||1922818 ||6454838 ||53388029 ||21598
|-
|- style="writing-mode: vertical-rl"
|'''22''' ||2|| 18|| 71|| 41|| 1398|| 204|| 2611|| 4128|| 17329|| 14684|| 12556|| 177336|| 22508|| 446333|| 302811|| 2885033|| 1922818|| 6454838|| 53388029|| 21598
|'''23''' ||4 ||98 ||10 ||679 ||890 ||368 ||1261 ||16126 ||44115 ||152006 ||190534 ||238989 ||70813 ||419435 ||2247711 ||12795738 ||45071 ||? ||? ||?
|-
|'''23''' ||2|| 4|| 98|| 10|| 679|| 890|| 368|| 1261|| 16126|| 44115|| 152006|| 190534|| 238989|| 70813|| 419435|| 2247711|| 12795738|| 45071 ||? ||? ||?
|}
|}


Revision as of 17:55, 21 July 2026

Walled Cities
x=64, y = 64, rule = B45678/S2345 ! #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 2345/45678
B45678/S2345
Rule integer 15608
Character Stable
Black/white reversal B01278/S5678

Walled Cities is a Life-like cellular automaton. Its rulestring is B45678/S2345, which means that a cell is born when it has from 4 to 8 alive neighbours, and an alive cell survives when it has from 2 to 5 alive neighbours.

Rectangle-shaped soups tend to generate a rectangular stable wall, which surrounds a region of high activity.

Notable patterns

Because the lowest birth condition in Walled Cities is B4, no pattern can escape its initial bounding box. There are no spaceships or infinitely growing objects.

Still lives

Non-existence of low birth conditions and existence of many low survival conditions allows for a large variety of still lives. There are at least 42 strict still lives with population up to 8; they are shown below, together with examples of bigger ones.

x = 152, y = 44, rule = B45678/S2345 2o$o3$2o5b2o5bo$2o4b2o5b3o3$8bo5bo$3o4b2o4b3o$2o4b2o6bo2$39b2o$8b2o4bo 3bo20bo$3o4b2o4b3o2b2o4b2o5b3o4bo$3o3b2o6b2o2b3o2b4o3b3o4b2o3$40b2o14b o$9bo6bo23bo14b2o24bo23bo$4o4b2o5b3o21bo14bo15bo9b2o22b2o$3o4b2o7b3o 19bo14bo14b2ob2o4b3o22b2o$6b2o29b2o13b2o14bo3bo5bo22b2o3$41b2o5b2o7bo 5b2o17bo6b2o15bo20b2o5b2o$9b2o5bo24bo6bo7b2o5b2o6bo9b2o6bo7bo7b2o6b2o 6bo5bo6bo$4o4b2o5b3o6b3o4b4o5bo6bo7bo6bo7bob2o6bo7bo7b3o5bo7b2o6b3o3bo 6bo7bo7bo$4o3b2o7b3o4b5o2b4o5bo7bo6bo6bo6b2o3bo3b3o5b3o5b3o6b2o6b2o6b 2o5b2o5b2o4b3ob2o3b2ob2o$6b2o9bo20bo7bo6bo6b2o6bo9bo7bo7bo6b2o6b2o6b2o 6b2o4b2o6bo3bo2b2o3bo$37b2o6b2o5b2o3$2b2o7bo7b3o4b2o3b2o3bo16b2o11b4o 9b5o22bo$bo2bo5bobo5bo3bo3b2o3b2o2b3o14bo2bo10b4o7b2o5b2o19b2o$o4bo3bo 3bo3bo5bo4bobo5b2o6bo5b2o4b2o7bo4bo6bo7bo18b2o$o4bo2bo5bo2bo5bo5bo8bo 4b3o3bo8bo5bo6bo4bo9bo16bo$bo2bo4bo3bo3bo5bo4bobo8bo3b2o3bo10bo2b2o8b 2o2bo9bo15bo$2b2o6bobo5bo3bo3b2o3b2o7bobo5bo10bo2b2o8b2o2bo9bo13b2o$ 11bo6b2ob2o3b2o3b2o8bo7bo8bo5bo6bo4bo9bo12b3o$50b2o4b2o7bo4bo5bo9bo7b 5o$52bo2bo10b4o7bo7bo7b5o$53b2o11b4o7b2o5b2o4b3o$79b5o5b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Oscillators

Any pattern that doesn't stabilize eventually turns into an oscillator. For large regions this can take a while, but because the space inside any wall is finite, it has finite configurations and must eventually become periodic.

Walled Cities allows for the existence of oscillators with the smallest possible rotor (and lowest possible heat), which can be attached to different stators:

x = 23, y = 8, rule = B45678/S2345 2o3b2o3b3o2b3o$obo2bobo2bobo2bobo$2o4b2o2b2o3b3o2$22bo$2b2o4b2o4b3o4b 2o$bobo3bobo3bobo4bobo$3o3b4o2b3o4b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Here are some small oscillators with periods from 2 to 39.[1]

x = 96, y = 38, rule = B45678/S2345 11bo10bo8b3o7b3o8b2o7b4o6b5o5b4o6b5o$10bobo8bobo6bo3bo5b2obo7bo2bo5b2o b3o4bo3b2o4b2o2b2o4bo4bo$10b3o7bo3bo6bobo6b4o6bo3bo5b2o2bo5bob4o4bob4o 4bo4bo$21bobo8bo7b4o7bo2bo5b4o6b6o5b5o4b2o3bo$22bo18b2o9b2o17b5o6b3o6b 2o2bo$92b3o5$b4o6b5o4b4o6b6o6b3o6b4o8bo6b6o15b4o$o4bo4b2obobo4bo2b2o5b o2b4o4b2o2bo4b6o6b3o5bob4o14bo2b3o$o2bobo4b2obobo4bobo2bo4bo2b3o4bo2bo bo4bo2bobo5bo2b2o4b6o14b3ob2o$bo3bo4b6o4bob4o4b2o2bo5b4obo5b2o2bo4b2o 4bo4bo2bo15b6o$2bo2bo14b5o5b4o7b5o6b3o6b2o3bo5b2o17b4o$3b2o57bo2bo$63b 2o4$b5o14b4o$o3b2o14bo2b2o$4obo14bo2b3o$b4o15bo2b3o$2b2o17b4o6$4bo7b3o 57b3o6b3o$3b3o5bob3o55bo3bo4bo3bo$2bob2o4b3ob2o54bo5bo3bo4bo$bo3bo4b6o 54bobo3bo3b3o2bo$o4bo4b6o54bo2bo2bo3b5o$o4bo65bo3bo4b4o$bo2bo67b3o$2b 2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

A typical city usually fills in the entire area it can reach, that is, the intersection of its bounding box and its bounding diamond.

x = 21, y = 23, rule = B45678/S2345 16b2o$16bo$16bo$15bo$15bo$o13bo$o13bo$6o7bo$6b6obo$12b2o$12bobo$12bob o$11bo3bo$11bo4b2o$10bo7bo$10bo8b2o$9bo9b2o$9bo$8bo$7bo$7bo$5b2o$6bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 6 LOOP 60 ]]
(click above to open LifeViewer)

This happens for majority of apgsearch-size soups, so simple objects in this rule are very difficult to find using apgsearch.

Periods of oscillators emerging from hollow rectangles are an extensively researched topic in Walled Cities.[2] A 3 × n rectangle always results in a p2 oscillator, with every cell inside flashing on in one phase and flashing off in the other phase. Otherwise, if a rectangle has sides m × n, it will produce an oscillator with period listed in the table:

m
n 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
4 2
5 4 3
6 2 2 5
7 4 9 4 7
8 4 7 14 7 7
9 4 5 5 11 17 10
10 6 8 14 28 11 10 17
11 4 10 7 2 10 20 56 17
12 4 11 19 40 50 41 24 3 48
13 4 6 21 32 11 58 222 69 716 56
14 10 26 18 51 18 118 384 31 176 504 31
15 4 22 46 2 48 169 219 748 2004 1041 453 58
16 4 66 46 54 69 176 85 461 2073 2412 2979 9811 57
17 4 54 32 185 88 46 468 262 220 4746 1612 11433 62146 518
18 14 64 28 4 199 47 373 164 2241 3158 24243 521 63817 54080 677
19 4 85 18 79 33 353 101 3127 2106 10938 4270 57784 117911 52951 385624 674
20 4 20 18 208 404 512 339 6784 14769 13819 2388 46276 218102 594883 346159 132674 5160
21 4 77 19 820 486 696 1518 6519 5875 8649 105386 41636 219190 380124 1601041 4675530 1513062 22118
22 18 71 41 1398 204 2611 4128 17329 14684 12556 177336 22508 446333 302811 2885033 1922818 6454838 53388029 21598
23 4 98 10 679 890 368 1261 16126 44115 152006 190534 238989 70813 419435 2247711 12795738 45071 ? ? ?

Periods of oscillators emerging from hollow squares (hypotenuse of the triangle above) are listed as sequence A178880 in OEIS.[3] Sequence starts with two 1's, for the 1 × 1 dot (which disappears, leaving unchanging void) and 2 × 2 block, a still life.

See also

References

  1. ↑ Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
  2. ↑ Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
  3. ↑ https://oeis.org/A178880
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