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A cross for the predecessor of a city
for now I'm done
Tag: Replaced
 
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{{Rule
nothing
|name            = Walled Cities
|char            = Stable
|b              = 45678
|s              = 2345
|reversal        =
|ruleinteger    =
}}
'''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 40 strict still lives with population up to 8; they are shown below, together with examples of bigger ones.
{{EmbedViewer
|rle = x = 141, y = 44, rule = B45678/S2345
2o$o3$2o5b2o5bo$2o4b2o5b3o3$8bo5bo$3o4b2o4b3o$2o4b2o6bo2$35b2o$8b2o4bo
20bo$3o4b2o4b3o4b2o5b3o4bo$3o3b2o6b2o3b4o3b3o4b2o3$36b2o7bo$9bo4bo21bo
7b2o24bo23bo$4o4b2o3b3o19bo7bo15bo9b2o22b2o$3o4b2o5b3o17bo7bo14b2ob2o
4b3o22b2o$6b2o25b2o6b2o14bo3bo5bo22b2o3$37b2o7bo5b2o17bo6b2o15bo20b2o
5b2o$9b2o3bo22bo7b2o5b2o6bo9b2o6bo7bo7b2o6b2o6bo5bo6bo$4o4b2o3b3o4b3o
4b4o5bo7bo6bo7bob2o6bo7bo7b3o5bo7b2o6b3o3bo6bo7bo7bo$4o3b2o5b3o2b5o2b
4o5bo7bo6bo6b2o3bo3b3o5b3o5b3o6b2o6b2o6b2o5b2o5b2o4b3ob2o3b2ob2o$6b2o
7bo18bo7bo6b2o6bo9bo7bo7bo6b2o6b2o6b2o6b2o4b2o6bo3bo2b2o3bo$33b2o6b2o
3$2b2o7bo7b3o4b2o3b2o3bo16b2o11b4o9b5o22bo$bo2bo5bobo5bo3bo3b2o3b2o2b
3o14bo2bo10b4o7b2o5b2o19b2o$o4bo3bo3bo3bo5bo4bobo5b2o6bo5b2o4b2o7bo4bo
6bo7bo18b2o$o4bo2bo5bo2bo5bo5bo8bo4b3o3bo8bo5bo6bo4bo9bo16bo$bo2bo4bo
3bo3bo5bo4bobo8bo3b2o3bo10bo2b2o8b2o2bo9bo15bo$2b2o6bobo5bo3bo3b2o3b2o
7bobo5bo10bo2b2o8b2o2bo9bo13b2o$11bo6b2ob2o3b2o3b2o8bo7bo8bo5bo6bo4bo
9bo12b3o$50b2o4b2o7bo4bo5bo9bo7b5o$52bo2bo10b4o7bo7bo7b5o$53b2o11b4o7b
2o5b2o4b3o$79b5o5b3o!
|position = center}}
 
===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:
 
{{EmbedViewer
|rle = x = 23, y = 8, rule = B45678/S2345
2o3b2o3b3o2b3o$obo2bobo2bobo2bobo$2o4b2o2b2o3b3o2$22bo$2b2o4b2o4b3o4b
2o$bobo3bobo3bobo4bobo$3o3b4o2b3o4b3o!
|position = center}}
 
A typical city usually fills in the entire area it can reach, that is, the intersection of its [[bounding box]] and its [[bounding diamond]].
 
{{EmbedViewer
|rle = 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!
|viewerconfig = #C [[ AUTOSTART GPS 6 LOOP 60 ]]
|position = center}}
 
Periods of oscillators emerging from hollow rectangles are an extensively researched topic in this rule. If a rectangle has sides {{times|m|n}}, it will produce an oscillator with period listed in the table:
 
{|class = "wikitable"
|-
! !!colspan="22"|m
|-
!n ||3 ||4 ||5 ||6 ||7 ||8 ||9 ||10 ||11 ||12 ||13 ||14 ||15 ||16 ||17 ||18 ||19 ||20 ||21 ||22 ||23
|-
|'''3''' ||2
|-
|'''4''' ||2 ||2
|-
|'''5''' ||2 ||4 ||3
|-
|'''6''' ||2|| 2|| 2|| 5
|-
|'''7''' ||2|| 4|| 9|| 4|| 7
|-
|'''8''' ||2|| 4|| 7|| 14|| 7|| 7
|-
|'''9''' ||2|| 4|| 5|| 5|| 11|| 17|| 10
|-
|'''10''' ||2|| 6|| 8|| 14|| 28|| 11|| 10|| 17
|-
|'''11''' ||2|| 4|| 10|| 7|| 2|| 10|| 20|| 56|| 17
|-
|'''12''' ||2|| 4|| 11|| 19|| 40|| 50|| 41|| 24|| 3|| 48
|-
|'''13''' ||2|| 4|| 6|| 21|| 32|| 11|| 58|| 222|| 69|| 716|| 56
|-
|'''14''' ||2|| 10|| 26|| 18|| 51|| 18|| 118|| 384|| 31|| 176|| 504|| 31
|-
|'''15''' ||2|| 4|| 22|| 46|| 2|| 48|| 169|| 219|| 748|| 2004|| 1041|| 453|| 58
|-
|'''16''' ||2|| 4|| 66|| 46|| 54|| 69|| 176|| 85|| 461|| 2073|| 2412|| 2979|| 9811|| 57
|-
|'''17''' ||2|| 4|| 54|| 32|| 185|| 88|| 46|| 468|| 262|| 220|| 4746|| 1612|| 11433|| 62146|| 518
|-
|'''18''' ||2|| 14|| 64|| 28|| 4|| 199|| 47|| 373|| 164|| 2241|| 3158|| 24243|| 521|| 63817|| 54080|| 677
|-
|'''19''' ||2|| 4|| 85|| 18|| 79|| 33|| 353|| 101|| 3127|| 2106|| 10938|| 4270|| 57784|| 117911|| 52951|| 385624|| 674
|-
|'''20''' ||2|| 4|| 20|| 18|| 208|| 404|| 512|| 339|| 6784|| 14769|| 13819|| 2388|| 46276|| 218102|| 594883|| 346159|| 132674|| 5160
|-
|'''21''' ||2|| 4|| 77|| 19|| 820|| 486|| 696|| 1518|| 6519|| 5875|| 8649|| 105386|| 41636|| 219190|| 380124|| 1601041|| 4675530|| 1513062|| 22118
|-
|'''22''' ||2|| 18|| 71|| 41|| 1398|| 204|| 2611|| 4128|| 17329|| 14684|| 12556|| 177336|| 22508|| 446333|| 302811|| 2885033|| 1922818|| 6454838|| 53388029|| 21598
|-
|'''23''' ||2|| 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. Sequence starts with two 1's, for the {{times|1|1}} dot (which disappears, leaving unchanging void) and {{times|2|2}} block, a still life.
 
==See also==
*{{rl|Vote}}
*{{rl|Rings 'n' Slugs}}
 
==References==
<references/>
 
==External links==
 
 
[[Category:Life-like cellular automata]]

Latest revision as of 10:55, 3 August 2026

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