User:Melwin22/Workspace: Difference between revisions

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===Still lives===
===Still lives===
Some possible still lives, with population up to 9, are shown below.
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
{{EmbedViewer
|rle = x = 70, y = 36, rule = B45678/S2345
|rle = x = 141, y = 44, rule = B45678/S2345
2o$o3$2o5b2o5bo$2o4b2o5b3o3$8bo5bo$3o4b2o4b3o$2o4b2o6bo3$8b2o4bo$3o4b
2o$o3$2o5b2o5bo$2o4b2o5b3o3$8bo5bo$3o4b2o4b3o$2o4b2o6bo2$35b2o$8b2o4bo
2o4b3o4b2o6b3o$3o3b2o6b2o3b4o4b3o2$38bo$9bo4bo22b2o$4o4b2o3b3o20bo$3o
20bo$3o4b2o4b3o4b2o5b3o4bo$3o3b2o6b2o3b4o3b3o4b2o3$36b2o7bo$9bo4bo21bo
4b2o5b3o18bo$6b2o26b2o$46bo$37b2o6b2o$4o5b2o3bo5b3o5b4o5b2o5bo$4o4b2o
7b2o24bo23bo$4o4b2o3b3o19bo7bo15bo9b2o22b2o$3o4b2o5b3o17bo7bo14b2ob2o
3b3o3b5o3b4o5bo6bo$7b2o5b3o18bo6bo$6b2o7bo18b2o5b2o2$10bo27bo6b2o5b2o
4b3o22b2o$6b2o25b2o6b2o14bo3bo5bo22b2o3$37b2o7bo5b2o17bo6b2o15bo20b2o
7bo7bo$5o4b2o26b2o6bo6b2o6b2o6b2o$4o4b2o26bo7bo6bo7bo7bo$7b2o26bo7bo5b
5b2o$9b2o3bo22bo7b2o5b2o6bo9b2o6bo7bo7b2o6b2o6bo5bo6bo$4o4b2o3b3o4b3o
2o7bo7bo$6b2o26b2o6b2o5b2o5b2o7bo$34b2o6b2o12b2o6bo$63b2o!
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}}
|position = center}}


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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 lowest possible [[heat]], equaling 1:
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
{{EmbedViewer
|rle = x = 18, y = 3, rule = B45678/S2345
|rle = x = 23, y = 8, rule = B45678/S2345
2o3b2o3b3o2b3o$obo2bobo2bobo2bobo$2o4b2o2b2o3b3o!
2o3b2o3b3o2b3o$obo2bobo2bobo2bobo$2o4b2o2b2o3b3o2$22bo$2b2o4b2o4b3o4b
2o$bobo3bobo3bobo4bobo$3o3b4o2b3o4b3o!
|position = center}}
|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 = 24, y = 27, rule = B45678/S2345
18b2o$17bobo$17bobo$17bobo$17bobo$18b2o$9b5o2b3o$7b2o4b3o2bo$7bo2b3o2b
2obo$6b17o$11o3b2o2b2o3bo$3b2ob4ob4o2b2o4bo$4b11ob3o3bo$4bo2b5o2b9o$3b
o4bobob6o4bo$3bo4b2ob2o2b3o4bo$4bo2b5o2bob2o4b2o$4bobo2b3o2bob2o4bo$4b
obobob2o2b2o6bo$5b2obobob4o5b2o$6b8ob2o2b2obo$7b15o$10bo4bo$8b4o2b2o$
6b2o3bo2bo$5bo5bobo$12bo!
|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==
==See also==

Revision as of 17:55, 17 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
Character Stable

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.

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! #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)

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 = 24, y = 27, rule = B45678/S2345 18b2o$17bobo$17bobo$17bobo$17bobo$18b2o$9b5o2b3o$7b2o4b3o2bo$7bo2b3o2b 2obo$6b17o$11o3b2o2b2o3bo$3b2ob4ob4o2b2o4bo$4b11ob3o3bo$4bo2b5o2b9o$3b o4bobob6o4bo$3bo4b2ob2o2b3o4bo$4bo2b5o2bob2o4b2o$4bobo2b3o2bob2o4bo$4b obobob2o2b2o6bo$5b2obobob4o5b2o$6b8ob2o2b2obo$7b15o$10bo4bo$8b4o2b2o$ 6b2o3bo2bo$5bo5bobo$12bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)

Periods of oscillators emerging from hollow rectangles are an extensively researched topic in this rule. If a rectangle has sides m × n, it will produce an oscillator with period listed in the table:

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 1 × 1 dot (which disappears, leaving unchanging void) and 2 × 2 block, a still life.

See also

References


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