OCA:Walled Cities

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 54 strict still lives with population up to 8; they are shown below, together with examples of bigger ones.

x = 229, y = 44, rule = B45678/S2345 2o$o3$2o5b2o5bo$2o4b2o5b3o3$8bo5bo$3o4b2o4b3o$2o4b2o6bo2$39b2o$8b2o4bo 3bo20bo$3o4b2o4b3o2b2o4b2o5b3o4bo$3o3b2o6b2o2b3o2b4o3b3o4b2o3$40b2o14b o13b2o13bo$9bo6bo23bo14b2o13bo14b2o47bo31bo$4o4b2o5b3o21bo14bo14bo14bo 7bo40b2o30b2o$3o4b2o7b3o19bo14bo13b2o13b2o6b2ob2o35b3o30b2o$6b2o29b2o 13b2o14bo14bo6bo3bo36bo30b2o3$41b2o5b2o7bo5b2o6b2o5b2o6bo5b2o7bo7bo7bo 9bo7bo6bo7b2o15bo6bo7b2o20b2o5b2o$9b2o5bo24bo6bo7b2o5b2o6bo6b2o6b2o2b 2o2b2o4bob2o4bob2o4bob2o4bob2o6b2o6b2o6bo7bo7b2o6b2o6bo7b2o6bo5bo6bo$ 4o4b2o5b3o6b3o4b4o5bo6bo7bo6bo7bo6bo7bo4bo4bo2b2o3bo2b2o3bo3bo3bo2b2ob o7bo7bo7bo7b3o5bo7bo7bo7b2o6b3o3bo6bo7bo7bo$4o3b2o7b3o4b5o2b4o5bo7bo6b o6bo7bo5b2o7bo13bo8bo6b2o6bo7b3o5b3o5b3o5b3o6b2o6b2o6b2o6b2o6b2o5b2o5b 2o4b3ob2o3b2ob2o$6b2o9bo20bo7bo6bo6b2o5b2o7bo5b2o47bo7bo7bo7bo6b2o6b2o 6b2o6b2o6b2o6b2o4b2o6bo3bo2b2o3bo$37b2o6b2o5b2o14bo14bo3$2b2o7bo7b3o4b 2o3b2o3bo16b2o11b4o9b5o22bo$bo2bo5bobo5bo3bo3b2o3b2o2b3o14bo2bo10b4o7b 2o5b2o19b2o$o4bo3bo3bo3bo5bo4bobo5b2o6bo5b2o4b2o7bo4bo6bo7bo18b2o$o4bo 2bo5bo2bo5bo5bo8bo4b3o3bo8bo5bo6bo4bo9bo16bo$bo2bo4bo3bo3bo5bo4bobo8bo 3b2o3bo10bo2b2o8b2o2bo9bo15bo$2b2o6bobo5bo3bo3b2o3b2o7bobo5bo10bo2b2o 8b2o2bo9bo13b2o$11bo6b2ob2o3b2o3b2o8bo7bo8bo5bo6bo4bo9bo12b3o$50b2o4b 2o7bo4bo5bo9bo7b5o$52bo2bo10b4o7bo7bo7b5o$53b2o11b4o7b2o5b2o4b3o$79b5o 5b3o! #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.

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. 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 produces 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. Sequence starts with two 1's, for the 1 × 1 dot (which disappears, leaving unchanging void) and 2 × 2 block, a still life; then both 3 × 3 and 4 × 4 have period of 2, and so on.

See also

• Vote
• Rings 'n' Slugs

External links

• Walled Cities at Adam P. Goucher's Catagolue