User:Melwin22/Castles/random patterns

Diehards, methuselahs and other stuff.

Diehards

Not counting fuses, the longest-lasting object I found that leaves nothing behind is the pumpkin. It reaches a maximum population of 110 and dies after 596 generations. Almost any solid object with population above 100 turns into a proper castle, and exceptions like the pumpkin are very rare.

x = 6, y = 6, rule = B3678/S135678 3b2o$2b2o$6o$2o2b2o$6o$6o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

L	Name	BB	MCPS	I	F	L/I	F/I	F/L	L/MCPS
596	Pumpkin	6 × 6	26	26	0	22.9	0	0	22.9

Pentaselah

Five cells are enough to make something interesting happen. This tiny pattern becomes a large, symmetrical castle with no chambers. In generation 2 it has 7 live cells, but a smaller bounding box, only 3x3. In generation 3, it also has 5 live cells, and that was the original pentaselah I found.

x = 4, y = 4, rule = B3678/S135678 2bo$2b2o$o$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

L	Name	BB	MCPS	I	F	L/I	F/I	F/L	L/MCPS
6224	Pentaselah	4 × 4	6	5	48753	1244.8	9750.6	7.833	1037.3

F refers here to the population in generation 6224, when everything settles down into periodic oscillations. The largest population ever reached is 48779, and in its final form, pentaselah is technically an oscillator with period 7392, although this number is simply a multiple of periods of all wall defenders (32 for flag bearer, 11 for rubble, 7 for half-ballista, 3 for bird's nest).

Hexaselah

A similar pattern that also commonly appears in soups. This one has a minimum population of 6 cells, and similarly to the pentaselah, in generation 1 it has more cells than the starting population, but a 3x3 bounding box.

x = 4, y = 3, rule = B3678/S135678 o2bo$b3o$o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

As a methuselah it's not that impressive, but might still be useful due to its simplicity.

L	Name	BB	MCPS	I	F	L/I	F/I	F/L	L/MCPS
684	Hexaselah	4 × 4	6	6	1880	114	313.3	2.749	114

Pentaselah + other objects

Combined with other patterns, pentaselah can react in various ways, which can be used to synthesize more complex objects. For example, a reaction with a single duoplet can create a p24 pulsating eye or a p42 butterfly.

x = 44, y = 60, rule = B3678/S135678 30b2o$12bo17b2o$13bo19bo$15bo16b2obo$14b2obo$33bo$15bo4$bo$2o$3bo34bo$ 2bo34b2o$23b2o15bo$21bo17bo$20b2obo$8bo$7bo13bo2bo14b2o$23bo5$37bo$19b o16b2o$bo16b2o4bo14bo$2o19bo3bo12bo$3bo16bo$2bo4bo$7bo$6bo12bo$20bo17b 2o$38b2o3$6bo$5b2o13bo$8bo10b2o14b2o$7bo14bo$21bo16bo3bo$37b2obobo2$ 20b2o16bo$5bo$4bo4$bo32bo$2o31b2o$3bo32bo$2bo16bo15bo$18b2obo$9bo13bo$ 8bo10bo2bo2$6bo$5bo36bo$43bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

More of such reactions are shown here.

Monstrosity

Castles are not an explosive rule, although sometimes it really looks like the chaotic growth doesn't end. The best example of such growth is what I call the "monstrosity". Perturbing evolution of a pentaselah in just the right way results in a greatly extended lifespan and greatly embiggened final pattern.

x = 10, y = 13, rule = B3678/S135678 9bo$9bo8$2bo$2b2o$o$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]

(click above to open LifeViewer)

L	Name	BB	MCPS	I	F	L/I	F/I	F/L	L/MCPS
84984	Monstrosity	10 × 13	15	7	2578423	12140.6	368346.1	30.34	5665.6

Again, F refers to the population in generation 84984, when everything settles down into periodic oscillations. The largest population ever reached is 2578474.

Monstrosity has a total of 927 wall defenders (see the census) and 59 chambers, ranging from anti-blocks (14 instances) and anti-ponds (7 instances) to the enormous central chamber, which has an area of over 107k cells. This chamber is over twice as big as the original pentuselah, but unfortunately there doesn't seem to be a way to fit it inside.

Evolution of the monstrosity takes so long that it's useful to write out some interesting phases.

Generation	Description
0	Smallest MCPS of 15, smallest population of 7 (equaled in generation 3), smallest bounding box of 10 × 13 (equaled in gen 1, 2, 4, 8, 9 and 10),
28	Population falls below the generation number, and stays below until generation 130.
118	Population reaches 100.
176	Only time when population is equal to the generation number.
179	Last time when population is smaller than generation number.
369	Pentaselah sequence hits the domino and stops being symmetrical.
705	Population reaches 1k.
726	Last generation in which there is an empty cell inside the original 10 × 13 bounding box.
1046	Final population of the hexaselah is reached. Monstrosity took 362 more generations than hexaselah to get here, but from this point, growth really starts speeding up.
2241	Population reaches 10k.
2897	First chamber forms.
5050	Final population of the unaltered pentaselah is reached, 1171 generations earlier than normal.
7365	Population reaches 100k. At this point the pattern grows in almost all directions and looks very similar to a typical blob in explosive rules.
12476	Third chamber forms, much bigger than previous two.
28741	Southern side of the pattern stabilizes.
30935	A giant chamber, ultimately second in terms of area, is formed.
31842	Population reaches 1M.
32974	A diagonal wickstretcher appears.
34363	Wickstretcher is destroyed.
≈37420	A different kind of a diagonal wickstretcher appears.
≈39230	Second wickstretcher is destroyed.
49544	Main chamber is closed off.
53307	Main chamber stabilizes. Growth now happens only on the northern side of the pattern.
60190	Population reaches 2M.
82193	North-east corner stabilizes.
84984	North-west corner stabilizes.

Integer constructions

Most small integer constructions in Castles disappear quickly or turn into dominoes/duoplets/blocks/simple oscillators. Here are all the exceptions up to 300 and some more examples up to 2000:

Integer	Description
31	Creates the p24 pulsating eye.
39	Turns into gen 1 of a hexaselah after 12 generations.
103	Forms a miniature castle with two bird nests and one small archer.
205	Forms a miniature castle with two guards.
215	Creates a p12 oscillator.
252	Forms the largest castle out of all sub-300 integers (final population 322k). This is quite a significant result for a pattern with C2 symmetry.
256	Forms a castle with final population of 206k.
266	Forms a castle with final population of 113k.
293	Forms a mid-sized castle with 9 wall defenders.
344	Creates a very rare 8-cell still life.
395	Forms the largest castle out of all sub-1000 integers (final population 2.153M) with tons of chambers. Longer lifespan and smaller bounding box than the monstrosity, although monstrosity still wins on final population and four main metrics.
452	A die hard that lasts 293 generations before disappearing.
712	Turns into gen 2 of a pentaselah at generation 34.
744	Forms a large castle, which emits a diagonal wickstretcher (smallest integer construction that exhibits infinite growth).
795	Forms a castle that also emits the same diagonal wickstretcher.
854	Turns into gen 4 of a pentaselah after 56 generations.
926	Forms a castle that emits two diagonal wickstretchers.
1029	A die hard that lasts 417 generations before disappearing.
1276	Forms the largest castle out of all sub-2000 integers (final population 2.323M, still smaller than the monstrosity).
1296	Creates a very rare 8-cell still life (different from the 344 integer).
1475	Creates a rare p2 oscillator.
1675	A die hard that lasts 471 generations before disappearing.
1954	Creates a rare p2 oscillator (different from the 1475 integer).

A comparison between the most notable integer methuselahs. In each case, the "starting" pattern is considered to be the integer moved forward to some Nth phase, where it has minimal I.

L	Name	BB	MCPS	I	F	L/I	F/I	F/L	L/MCPS
17190	252 (+ 1 tick)	13 × 7	34	28	322557	613.9	11519.9	18.764	505.6
118949	395 (+ 2 ticks)	11 × 7	31	29	2153950	4101.7	74274.1	18.108	3837.1
99163	1276 (+ 13 ticks)	12 × 7	19	16	2323348	6197.7	145209.3	23.43	5219.1

Repunit numbers quickly disappear/stabilize for digits 0, 1, 2, 3, 5, 7 and 8; castle forming occurs only for 4, 6 and 9.