Spaceships in R3 Cross rules (R3, N+)
Posted: July 9th, 2024, 8:45 am
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:
but in rules without S2 or S3 the top alive cell in the middle dies, forming:
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:
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??? ???????
??????? ???????
Code: Select all
....... ...o...
...o... .......
...o... -> ???????
???o??? ???????
??????? ???????
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!