Assimilation (B345/S4567)

[137ben]

I have been searching for interesting patterns in this rule.

Most patterns become oscillators of p2,p3,p4,p6, or p12. But interestingly, the blockick CA that it emulates (in the same way as 2x2) behaves very differently. I have been testing this by generating a random soup, and then replacing each live cell with a block and each dead cell with a 2x2 square of dead cells (one of the scripts in the "golly scripts" thread can do this automatically). This naturally produces oscillators of extremely high periods. Similar to the rule "walled cities", (B45678/S2345), a wall of 2 by 2 blocks surrounds a jumble of pseudorandom 2x2 blocks in the center.

It is not known if assimilation allows for any spaceships. This is one problem I would really like to see solved: prove the existence, or lack thereof, of a spaceship in B345/S4567.

There are a few other challenges I am going to raise:

1. What is the highest period that can be obtained from a starting pattern which fits in a 16 by 16 box? (Note: the oscillator need not fit in a 16 by 16 box, but there must be an ustable pattern which DOES fit that is a predecessor of that oscillator).

2.Now try to find a pattern with the same requirements as challenge 1, but it cannot be made entirely of 2x2 blocks. How high of a period can now be obtained? As far as I have found, patterns made of 2x2 blocks tend to have much higher periods than other oscillators, but even non-blockic oscillators can have high periods.

3.It seems likely that arbitrarily high period oscillators are possible. I am interested to see a prove of this, even if it does not involve an algorithm to generate such high periods (such as a relay of spaceships). This is more difficult to prove than in life, because there are no naturally occurring spaceships.