The family tree of a pattern's cells

Docsy · Post by **Docsy** » December 29th, 2025, 3:46 am

I've recently been considering the idea of a cell's family tree. The idea is that a living cell can be considered to have given rise to a cell in the next generation if there's a living cell in the same position in the next generation, or if a cell neighboring it was born. These cells are its children, and it is the parent. Furthermore, the children of children of ... of children of a cell are descendants of that cell, and the original cell is the ancestor of those descendants.

B3/S23Super can actually simulate the descendancy of a cell, as "on no-trail 1" cells will always give birth to themselves when all alive cells have no trail. Using that principle, here's the descendants of every living cell in the glider in each distinct phase:

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x = 23, y = 8, rule = B3/S23Super
M2S2.SMS2.2SM2.3S2.3S$2.S4.S4.S4.M4.S$.S4.S4.S4.S4.M3$.M4.S4.S4.S4.S$
.2S3.MS3.SM3.2S3.2S$S.S2.S.S2.S.S2.S.M2.M.S!

As you can see, one cell in each phase doesn't ever actually take over the pattern, instead having only one child every time step. This is actually typical for spaceships from what I can tell, where in order for every cell of the spaceship to be colored white, cells in certain subregions have to have at least one of their cells be colored white to turn white, while coloring cells outside those subregions will never color the whole pattern on its own.

A curious pattern I noticed is that for spaceships of speed c/2, these subregions appear to be highly restricted in form, and I've yet to find a counterexample. Each one takes the shape of a period-2 subpattern that alternates between a t-tetromino shape and a line of 3 cells. This isn't trivially true, as there's OT cellular automata with speed limit c/2 that don't follow this for some spaceships. I get the sense that these subregions are in a sense the "engines" of the spaceship. There might be some sense that all c/2 spaceships have the same growth pattern underlying them. Apologies if this is not a novel observation, I couldn't find it mentioned before but I might lack the necessary terminology. Visualization:

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x = 63, y = 70, rule = B3/S23Super
33.M.M$33.M2.M$36.2M$38.M$36.4M$35.M4.S15.2M$37.M2.S9.2M3.M2.S$37.M2.
S14.M2.S$39.M14.2M.MS$35.4M17.M$31.M$30.M.2M2.M.S15.2M4.M$33.M4.2S16.
M2.M.S$37.MS14.M3.M3.2S$31.7M16.M.5MS$34.M.M18.2M3.M$31.M2.M$34.M.M18.
2M3.M$31.7M16.M.5MS$37.MS14.M3.M3.2S$33.M4.2S16.M2.M.S$30.M.2M2.M.S15.
2M4.M$.3MS4.4MS4.5MS7.M$M3.S3.M4.S3.M5.S11.4M17.M$4.S8.S9.S15.M14.2M.
MS$M2.M4.M3.M4.M4.M14.M2.S14.M2.S$10.M8.2M16.M2.S9.2M3.M2.S$35.M4.S15.
2M$36.4M$.4S4.4SM5.5S14.M$S3.S3.S4.S4.S4.M12.2M$4.M8.S9.S9.M2.M$S2.S4.
S3.S5.S3.S10.M.M$10.S9.S3$.2M6.3M6.4M$3MS4.4MS4.5MS10.S.S$2M.2S3.3M.2S
3.4M.2S9.S2.S$2.MS7.MS8.MS13.2S$38.S$36.4S$35.S4.M15.2S$.2S6.3S6.4S15.
S2.S9.2S3.S2.M$4S4.5S4.6S14.S2.S14.S2.S$2S.SM3.3S.MS3.4S.2S15.S14.2S.
2S$2.2S7.2S8.SM12.4S17.S$31.S$30.S.2S2.S.S15.2S4.S$33.S4.SM16.S2.S.S$
37.2S14.S3.S3.SM$31.7S16.S.6S$34.S.S18.2S3.S$31.S2.S$34.S.S18.2S3.S$31.
7S16.S.6S$37.2S14.S3.S3.MS$33.S4.MS16.S2.S.S$30.S.2S2.S.S15.2S4.S$31.
S$35.4S17.S$39.S14.2S.2S$37.S2.S14.S2.M$37.S2.M9.2S3.S2.S$35.S4.S15.2S
$36.4S$38.S$36.2S$33.S2.S$33.S.S!

Note that some of the cyan cells in some of the demonstrations are not actually part of the subregions that color the rest of the spaceship, and thus don't obey the pattern mentioned. They're just cyan because they happen to not have any white-colored ancestors. The more complicated spaceships have more than one of these regions, and that means that you have to color at least 1 cell in each to color the whole spaceship.

Also, I'd like to mention that while many c/4d spaceships have identical ancestor cells as well, essentially the front 4 cells of a glider, Enterprise shows that if the c/2 pattern is real, it doesn't have an extension to c/4d spaceships.