User:AforAmpere/SpeedLimits
This is a test page for writing up speed limit proofs for a future page in the OCA namespace.
All ships are oriented between north and northeast. We use the coordinate system with x going east and y going north. (k,l)c/p for k>=l is pointed mostly upwards. This (2,1)c/3 serves as an example:
| (2,1)c/3 oriented in the direction that will be used throughout the page. (click above to open LifeViewer) |
For future diagrams, state 0 will indicate cells forced to be off, state 1 cells forced to be on, state 2 cells uncertain, state 3 important marker on cells, and state 4 important marker off cells.
Non-B0 range 1 Moore
k+l<=p speed limit
For a given pattern, let C(t) := max(x + y) in generation t. C(t+1) < C(t)+2 as can be seen below. The only birth conditions that the red cell could theoretically be born on are B0 and B1c. We have excluded B0 for this section and B1c always causes patterns to expand in all directions. This means the red cell cannot be born on the next step, so C(t+1) < C(t) + 2. Thus, a pattern can only increase its maximum value for x + y by at most 1 every generation. So k+l <= p.
| Cell that needs to turn on highlighted in state 4. (click above to open LifeViewer) |