ConwayLife.com

This rule is Transers 2 with one more state (see here). Some of the patterns survive the transition from Transers 2 to Transers 3 including spaceships with the three velocities c, c/2, and 2c/3. The square p12 oscillator becomes p14. Explosive growth is subdued considerably.

But then I found this very unique dual velocity multi-branch qudratic gun (hard to describe): At first I didn't notice new branch locations were being created, but collisions between spaceships of different velocities create new dots that form a new branch location. The dots get cleared and allow another pair of spaceships with different velocities to pass through before a new set of dots reform at the same location. The branches act as pulse dividers, so the new streams of spaceships quickly become very sparse as the growing gun expands two dimensionally.

This sawtooth produces growing diagonal lines: The puffer is growing and the origin of the diagonal lines is moving, but the spacing of the lines still looks logarithmic. This pattern may not be a sawtooth by strict definition. Also, the line spacing seems to get more even though it is still increasing.

...The population graph for that pseudo-sawtooth you posted looks quite a bit like a semicubical parabola. Perhaps because the pattern spawns one instance of constant infinite growth at ever-increasing intervals. (Along with the constant infinite growth generated otherwise.)

I checked the population growth of the first pattern in this thread, the multi-branch gun, and found the growth is apparently not quadratic like I thought. It is a sawtooth but the overall growth looks very linear. I don't know how to prove it one way or the other but it makes sense that really there is only only one source being split many ways, not entirely new streams formed by newly created guns.

knightlife wrote:I checked the population growth of the first pattern in this thread, the multi-branch gun, and found the growth is apparently not quadratic like I thought. It is a sawtooth but the overall growth looks very linear. I don't know how to prove it one way or the other but it makes sense that really there is only only one source being split many ways, not entirely new streams formed by newly created guns.

This is basically correct. The core gun collider produces width-4 photons at a constant rate, and every two of these generate a set of two c/2s sent counterclockwise and one width-7 photon sent clockwise. It's basically a slide gun.

There's some sort of an extra effect though from that the ships in the far-off side streams take increasingly longer time to fully form. Probably overall adding up to linear minus an exponential decay term?