User:H. H. P. M. P. Cole/The R3 Cross Rulespace
(Work-in-progress)
Intro
The R3 Cross Rulespace is interesting to say the least. The dynamics are vastly different from that of the Moore neighbourhood.
Notation
Notation will follow totalistic Moore notation, with a N+ at the end to signify that the rule is in the Cross neighbourhood. For example Factorio is B3/S2/N+.
For simplicity non-B0 rules are considered only.
Naive analysis
This step is exceedingly important as one needs to find the regions of stability.
- B1 and B2 cause runaway behaviour.
- Rules with only B4 and above do not have natural spaceships as patterns cannot escape their bounding boxes.
Hence only rules with B3 as lowest birth transition are considered. This creates a total of 2^22 rules.
Speed limits
With B3 the speed-limit is c orthogonal. However, having S1 or any combination of S456789XED (Where x, e and d stand for 10, 11 and 12 respectively) make reduced-speed c/1o spaceships an impossibility.
With B2 the speed limit is 2c orthogonal.
Diagonal speed limits are half of the orthogonal speed limit, but spaceships are unlikely to reach this limit as they would have to zigzag without pausing.
Guaranteed small SLs
S0 will always produce o! as a still-life.
S1 will always produce 2o!, obo!, and o2bo! as still-lifes.
On first examination, one might conclude S2 might always produce an 3o! and 2obo!. However there is also B3. If B3 is added, then S2 will produce 2o$2o, 2o2$2o, 2o3$2o, obo2$obo, obo3$obo, and o2bo3$o2bo.
One might think that B3/S3/N+ cannot produce a still-life. However, there is one known, and it is this:
Sadly it has 24 cells.
4-cell objects and below
The threshold of 4 cells is important as some B3 rules have many 4-cell objects (most of them have 18: rules with S2 have 40). This is assuming S1 is not included.
B3 always produces this set of oscillators (assuming S0 and S1 are not added):
If S2 is added, this would result in this:
S2 also produces the characteristic pattern of Factorio: the factorialship.
It is interesting to note that the factorialship is extinguished when S3 or B4 are added.
S0 creates the dot, o!
Adding S0 creates this set of 4-cell oscillators:
Adding S02 creates this:
Note that any rules with B3/S02/N+ are explosive.
The only oscillator below 4 cells in B3/S1/N+ is this:
All other less-than-or-equal-to-4-cell configurations in B3/S1/N+ turn into 2o!s, obo!s, o2bo!s, and conglomerations of these. Here are all the xs4s:
In summary, here are the less-than-or-equal-to-4-cell non-trivial object counts for the B3 non-explosive rules:
| S0 | S1 | S2 | S3 and above | |
|---|---|---|---|---|
| p1 | 1, 1-cell | 3, 2-cell, 12, 4-cell | 6, 4-cell | none |
| p2 | 9, 4-cell | 1, 3-cell | 33, 4-cell | 18, 4-cell |
| c/1o | none | none | 1, 4-cell | none |
Non-phoenix oscillators
Phoenix oscillators are known for all B3 rules that do not have S1.
Non-phoenix oscillators are known for the below stable realms. it is not known whether other non-phoenix oscillators are known for other stable realms.
Smallest known for each:
B3/S0/N+ (also works in B3/S04/N+, B3/S05/N+, B3/S06/N+)
B3/S1/N+ (also works in B3/S16/N+)
B3/S2/N+
B3/S3/N+
B3/S4/N+
Stable realms with spaceships
There are two stable realms in the B3-space with natural spaceships. They are B3/S2/N+ and B3/S3/N+.
The first realm, B3/S2/N+, is the subject matter of this page: Factorio
Here is the spaceship:
When S3 is added, it generates a 4-cell and a 5-cell spaceship:
The rule is explosive, though.
The second realm, B3/S3/N+ is less interesting. There are no still-lifes, but there is still a common spaceship:
Both stable realms are extremely fragile and will often become explosive when one transition is added. Exceptions include:
Other stable realms which have spaceships include: B3/S1/N+ and B3/S16/N+ (below is a c/2):
B3/S0/N+ and its relatives have no known spaceships.
Other stable realms
Here is the list of all stable realms with the addition of B89XED/S789XED making the rule still stable (2048 rules each):
- B3/S/N+
- B3/S0/N+
- B3/S1/N+
- B3/S2/N+ (semi-stable)
- B3/S3/N+ (semi-stable)
- B3/S4/N+
- B3/S04/N+
- B3/S5/N+
- B3/S05/N+
With the addition of B89XED/S89XED (1024 rules each):
- B3/S6/N+
- B3/S06/N+
- B3/S16/N+
S7 often makes ash formation slower and density lower. The cause of this is unknown.
B789XED/S789XED causes large 'monoliths' which do not stabilize and grow bigger and bigger a la IceNine.
Adding S6789XED often causes large 'rockets' to form, which was the source of much exploration in the Cross rulespace until Factorio came along.