Spaceships in R3 Cross rules (R3, N+)

[H. H. P. M. P. Cole]

The R3 Cross rulespace is 2^26 rules large, 2^8 times larger than R1 OT Moore. However I think that this rulespace is fertile for ship searching.

First, some theorems regarding ships in R3 Cross rules without B0:

- With B1, all patterns expand at the speed of 3c orthogonal in all directions. Hence no finite patterns are possible.
- With the lowest birth condition being B4 or higher, there are no patterns whose bounding box can expand.
- The orthogonal speed limit of a rule with minimum birth condition B2 is 2c.
- The orthogonal speed limit of a rule with minimum birth condition B3 is c.
- Any rule with S0123 cannot have ships as every pattern's bounding box either stays the same or expands.

And another more complicated theorem:

- Any rule with B3 but no S2 or S3 cannot have c orthogonal spaceships.

Proof:


For a spaceship to be c/1o it should have the line-of-three frontend below:

Code: Select all

.......        ...o...
...o...        ...o...
...o...   ->   ???o???
???o???        ???????
???????        ???????

but in rules without S2 or S3 the top alive cell in the middle dies, forming:

Code: Select all

.......        ...o...
...o...        .......
...o...   ->   ???????
???o???        ???????
???????        ???????

and since higher-period c/1o ships must have a c/1o frontend (due to c/1o being the speed limit), then there are no c/1o ships in B3 rules without S2 or S3 (in R3 cross totalistic).
Hence it follows that the speed limit in this class of rules is c/2o since one cannot have the line-of-three frontend surviving for more than one generation.

Two open questions:
- Are there any other easily provable theorems about speed limits of ships in more complicated subspaces?
- Are there c/1o ships in every B3 rule which can support (at least) one?

To kick off this topic, three small c/1o ships that work in 2^18 rules each:

Code: Select all

# R3,C2,S2,B3,N+ - R3,C2,S0,2,4-12,B3,5-12,N+, 2^18
x = 6, y = 1, rule = R3,C2,S2,B3,N+
o2b3o!

Code: Select all

# R3,C2,S3,B3,N+ - R3,C2,S0,2-3,5-12,B3,5-12,N+, 2^18
x = 7, y = 1, rule = R3,C2,S3,B3,N+
o2b4o!

Code: Select all

# R3,C2,S3,B3,N+ - R3,C2,S0-1,3,5-12,B3,5-12,N+, 2^18
x = 6, y = 2, rule = R3,C2,S3,B3,N+
o2b3o$o2b3o!

[pifricted]

What’s the meaning of “2^18 rules?

[H. H. P. M. P. Cole]

Let me post my ship collections in assorted R3 cross rules here:

AlephAlpha found the c/3o, 2c/3o, c/4o, 2c/4o and c/2d in R3,C2,S2,B3,N+
wildmyron found the c/2o and c/3d ships in R3,C2,S2,B3,N+
C28 found the 4c/4o ship in R3,C2,S2,B3,N+

The rest of the ships were found by me using AlephAlpha's factoriosrc or apgsearch (in the case of R3,C2,S2,B3,N+).

c/1o, 2c/2o, 3c/3o, 4c/4o, 6c/6o, 8c/8o, 2c/3o, c/2o, 2c/4o, c/3o, c/4o, c/2d, c/3d

Code: Select all

x = 141, y = 59, rule = R3,C2,S2,B3,N+
o5bo6bo11bo3bo4bo4bo4bo4bo7bo12bo7b2o12bo8bo$o5bo6bo11bo3bo4bo4bo4bo4b
o7bo2bo9bo7bo9b2o9bobo17bo15b2o$o5bo6bo11bo3bo4bo4bo4bo4bo7bo20bo2bo7b
2o7b2o18bobo15bobo$59bo2bo6b3o5bo2bobo8bo29bo14bo2bo$18bo40bo2bo6bobo
15bo2bo8bo18bobo$o5bo6bo4bo6bo3bo4bo4bo4bo4bo9bobo6bo3bo5bo2bo6bo2bo6b
o19bo19b3o$17b3o7bo7bo2bo7b2o8b2o3bo8bo16bo11bo21bo18bo$7bo8bo18bo19b
o36bo8bo16b2o$7bo12bo17bo7b2o14bo7bo30bo17bo$7bo12bo6bo7bo8bo4bo6bo5b
o35bo2bo17bo$20bo6bo8bobo17bo30b2o9bo3bo17bo$7bo8bo20bo6bo4bo8b2o10bo
17bo13bo14bo$17bobo17b2o5bo4bo10b2o8bo17bo13bo15bo$35b2o8bo2bo13bo$61b
2o$36bo7bo4bo11bo35bo$37bo61bo$44bob2obo47bobo$36b2o$59b2o38b2o$36b2o
8bo13bo39bo$46bo13bobo$36bo10bo10bo$36b2o8bo52b2o$46b2o49bo$59b2o$37b
o59bo$34bo4bo19bo37bob2o$36bo2bo19bo$34bobo21bo40b2o$34bobo22b2o39bo$
34bobo21bobo38b2o$37bo20bo40bo$58bo41bobo2$102bo$99bo2bo$102bo$99b2ob
o$102bo2$99bo4$98b3o$98bobo$100bo$98bo4$98b2obo2$99b2o$98b2obo$98bo2b
o2$98bo!

c/1o, 2c/2o, 3c/3o, 4c/4o, 6c/6o, c/2o, 2c/4o, c/3o, c/2d

Code: Select all

x = 89, y = 51, rule = R3,C2,S3,B3,N+
o5bo7bo10bo19bo2bo10bo8bo16bo$o5bo7bo10bo8bo10bo2bo8b2obo7bo2bo12bobo
bo$o5bo7bo10bo8bo22b2obo8bo$o5b2o6bo10bo8bo10bo2bo10bo11bo11b2ob3o$5b
o26bobobo7bob2obo9bo9bo$7bo25bo23bo2bo23bobobo$o5bo7bo10bo6bo3bo8bo2b
o23bo10bobobo$26bo6bo10bo4bo7b2obo7bobo8b2obo$7bo16bo21b2o10bo9b2o2bo
10bobo$7bo16bo7bo2b2o7b2o2b2o7bo2bo23b2o$14bo3bo7bo5bo3bo18b2o3bo11bo
11bo$7bo10bo14bo10b2o2b2o11bo7bobo7b2o2b3o$14bo3bo5bo9bo22bobobo8bo11b
o$14bo3bo5bo32bobobo7b2o$24bo7bobo9bob2obo7bo2b2o$14bo29bo4bo11bo$14b
o3bo7bo7bo23bo2bo$15b3o26bob2obo6bo$46b2o9b3o10b2o$57b2o11bo$56bo3bo9b
o$68bo$58bo11bo2$70b2o$70b2o$70bo$71bo$71b2o$69bob2o$71b2o2$70bo$71bo
$69bo2bo2$72bo2$72bo$72bo$72bo3$72bo$72bo$71bo$68bo3bo$71bo$68bo$68bo
$68bo!

c/2o, 2c/4o, c/3o, c/2d

Code: Select all

x = 68, y = 32, rule = R3,C2,S1,B3,N+
2bobo6bo8bo6bobo$bo3bo5bo8bo$bo3bo5b2o6b2o5bo3bo22bo$11b2obo2bob2o5bo
3bo22bo2bob5o$2o3b2o19bobobo22bobo4bobo$o5bo20bobo21b2obob2ob2obo$60b
obo2bo$26bo3bo22bob2ob2o6bo$45bo14bobobo$26bo3bo16bo5bo12bo$26bo3bo15b
o5b2o$43bo2bobo2bo2bo7bo$42bo5bo3bo11bo$26bobobo13bo6b2o9bo3bo$44bo4b
o8bo$27bobo10b2o10bo2bob2o5b2o$27bobo19b2o2bo2bo3bo6bo$43bo15bo4bo$27b
obo10bo4bo8bo$26bo3bo24bo$25bobobobo11bobo7b2obo$27bobo13bobobo6b3ob2o
$26bo3bo9bo7bo8bo$26bo3bo8bob2o2bobo6b3ob3o$26bo3bo18bobo$47bo5b4o$24b
2o5b2o13bo2bobob2o2bo$26bo3bo16b2o4bo$53bo$25bo5bo15b2ob2o$25bo5bo$24b
o7bo!

c/2o, c/3o

Code: Select all

x = 16, y = 53, rule = R3,C2,S0,B3,N+
bo10bo$3bo10bo$o14bo$o10bobo$2o12b2o2$bo10bobo$3b2o$4b2o$13b2o$11b2o2b
o$bo$bo$4bo8bo$2bo10bo$13bo$bo8bo$bo13bo$15bo$2bobo$o3bo5bo$o11b2o$4b
o6b2o$12b3o$11bo2$14bo$11bo2$11bo$14bo2$11bo$12b2o$10b2o2$10bo$10bo$10b
2o2bo$12b2o$11bo$11bobo$12bo$11bo$12b2o$14bo2$10bo3bo$14bo2$12bo$14bo
$13bo!

c/1o, 2c/2o, 3c/3o, 4c/4o, 2c/3o, c/2o, 2c/4o, c/3o, c/2d

Code: Select all

x = 114, y = 37, rule = R3,C2,S2-3,B3,N+
o5bo6bo7bo5bo6b2o5b2o7bo6bo7bobo7bo27bo$o5bo6bo7bo5bo6b2obobob2o7bo6b
o8bo8bobo25bo$o5bo6bo7bo5bo9bobo25bobo9bo26b3obo$o11b3o35bob4obo19bo26b
o2b3obo$52b4o46bo3b2o$12bobo49bobobo8bo24bo7b2o$o5bo5bobo6bo2bo2bo36b
obobo8bo24bobo2bo3bo$12bobo22bobo24bob3o6bo33bo$12bobo9bo10bo5bo8bob4o
bo7bo41bobo2bo$24bo8bob2o3b2obo6bo6bo7bo9bo23bo7bo3bo$32b2ob3ob3ob2o5b
o6bo5bob4o8bo19bobo7bo2b4o$65bo3bo20bob6o4bo6b2o$32bo3b2ob2o3bo5bo6bo
7b2o2bo21b2o4b3o2b2ob5o$13bo10bo7bo3b2ob2o3bo5bo6bo5bo3b2o25bobo2b2o$
13bo23bobo24bo3bo6bo16bo2b3o4b2o$13bo19bo3bobo3bo19bo3bo7bobo10bo2bo12b
o$24bo7bo4bobo4bo44bo11bobo$23bobo7bo4bo4bo20bo2bo9bo5bo10b2o2bo4bo$13b
o9b3o6bo3bo3bo3bo18b2o12bo6b3o5b3o2bo$22bo3bo37bo18bo5bob2o3bo$24bo41b
o17b3obo2bo4b2o2bo2bo$23b4o9bobobo32bobo7bob3o3bo3b2o3b2o$22b2ob2o11b
o34bobo9bo2b2o4b2o3bobo$37bobo35bo11bo4b2o3b5o$36bo3bo42bobo8bo6b2o$35b
o5bo33bo9bobo3bo6bo$73bo11bo3b3o$73bo14bobobo$32bo2bo5bo2bo44b2o5bo$32b
o11bo30bo11bo8bo$32bo2bo2bo2bo2bo30bo17b4o$38bo34bo15bo4bo$32bo3bo3bo
3bo28bo$38bo$75bo$73bo$75bo!

c/1o, 2c/2o, 4c/4o, 2c/3o, c/2o, 2c/4o, c/3o, c/2d

Code: Select all

x = 76, y = 42, rule = R3,C2,S1,3,B3,N+
67b2o$2o5b2o5b2o8bo7bo10bobo6bo11b2obob2o$2o5b2o5b2o8bo10bo6bobo22bo2b
o$2o5b2o5b2o16bo3bo5bo3bo20bo3bo$46bo5bo17bo2bo$23bobo9bo6bobobo20bo4b
o$2o5b2o5b2o8bo10b2o5bo9bo15b3obobo$9bo62bobo$16bo9bo15bobo7bo22bo$9b
o6bo6bobo10bo5bo9b2o$23bobo18bo9bo19bo$6bo15bo3bo9bo17bo$6bo19bo9bo7b
o$26bo9bo18bo$35b2o7bo7bo2bo$26bo8bo$26bo25bo2bo$26bo$54bo$26bo17bo7b
o$44bo7bo$26bo17bo7bo2$52bobo$26bo27bo$26bo27bo3$54bo$54b2o3$54b2o$55b
o$54bo$55bo$54bo2$54bo3$54bo!

c/1o, 2c/2o, 3c/3o, 4c/4o, 2c/3o, c/2o, 2c/4o, c/3o

Code: Select all

x = 85, y = 47, rule = R3,C2,S1-2,B3,N+
o4bo7bo9bo9bo$o4bo7bo9bo9bo7bo5bo10bo8bo8bo7bo$2o2b2o6b3o8bo9bobo5bo2b
o2bo5bo4bo8bo8bo7bo$11bo3bo5b2obo19bo12bo$bo2bo6bo3bo19bo17bo12b3o$11b
o3bo5b4o10bo17bob2o9bobo$2o2b2o5bo3bo17bo31bo3bo$13bo7b2obo6bo10bo14b
o9bo8bobobobobo$o4bo14bo4bo5b2ob2o8bo12bo19b3ob3o$ob2obo36bo4bo19bo9b
2o3b2o$22bo9bobo7bo11bo3bo19b2ob2o$21b4o18bo10bo3bo19bo3bo$32bobo9bo9b
o3bo8bo10bobobo$23bo11bo6b2o34bobobo$21bob2o8bo10bo9bo2b2o$20bo4bo6b3o
22bo$22bo20b2o12bo18bo7bo$31bo22b3obo8bo8b2o5b2o$33bo8bobo20bobo9bo5b
o$31bobo7bobo2b2o17bobo9bo5bo$31bo9bo23bo$41bo5bo21bo$42bo3b2o19bobo$
41bo27bo9bobo$41bo$65bo11bobobobo2$44bo$44bo33bo3bo$78b2ob2o$44bo33bo
3bo5$79bobo$78bo3bo4$77b7o$77bo5bo$78bo3bo$78bo3bo$77bo5bo$78b2ob2o$77b
o2bo2bo!

c/3o (I was surprised that there was a ship in that rule)

Code: Select all

x = 8, y = 44, rule = R3,C2,S,B3,N+
3b3o$3b2o2bo$bo2b2o$2bo$6bo$bobo3bo3$3bo2$3bo$bo$3bo$bo2$bo4bo$4bo$3b
o2bo$2bo$3b3o2$2bo3bo$o$2bo$o3bo$bob2o$o$o5bo2$o2bo2bo$bo3bo$3bo2bo2$
2bo3bo2$3b2o2$2bobo4$2bobo$2b2o$4bo!

EDIT: added new ships in R3,C2,S1-2,B3,N+ and R3,C2,S1,3,B3,N+