User:Tropylium/LWOD variants
Subtraction of survival conditions
Interestingly, the removal of S0, S1, S2 on one hand, S8 on the other has minimal effect on the evolution of the rule.
Consider separating any pattern into three parts:
- the exterior: all empty cells rookwise connected to the vacuum outside the pattern
- the boundary: all live cells neighboring the exterior
- the interior: all other cells
For the five related rules missing any of the first three conditions, it can be proven by induction that many growing patterns will evolve in the exact same way:
- Assume that a pattern's boundary contains no cells that will die due to underpopulation.
- In the next generation, all newborn cells will have at least three neighbors.
- Therefore, no cells will die due to underpopulation in the next generation either.
Small, sparse patterns - whose boundary can contain dying cells - may however persist as oscillators; patterns such as phoenix 1 and various sparse agars known from Life function the same in some of these rules. Small spaceships are also, in principle, possible.
An open research question might be how long can a "flipper" area persist under growth, and whether "decaying shoots" can be engineered.
B3/D8 can also be proven to evolve in a very similar way to regular LWOD.
- Boundary cells must have at least one dead neighbor, and will therefore be never affected by the presence of D8.
- Since the boundary cannot shrink, the boundary in generation N+1 only depends on the boundary in generation N+1.
Therefore, any finite B3/D8 pattern will grow to fill in with ladders and irregular stable crystals exactly the same areas of the plane as the corresponding B3/D pattern; the only difference will be the texture of the interior.
Addition of birth conditions
Like all B34 rules, B34 Without Death grows in an octagon shape in all directions, usually (always eventually?) without leaving gaps.