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Adjustable-speed replicators

Posted: January 12th, 2022, 6:13 am
by muzik
This rule by bibunsekibun has an infinite family of 1D replicators with an infinite number of distinct speeds:

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x = 3, y = 1253, rule = B3aeiy4eqrty5cijn6ck7c8/S2-i3acein4cejkq5ir6cei7c8
Separating the two tubs causes the speed to approach c/2, but never reaches it exactly.

The size of the separation between the two tubs that allows for clean replication does not seem to grow in a regular fashion (3, 7, 12, 19, 24, 31, 36, 43, 48, ...)

It's also possible to time-offset to allow for replication as well. I haven't investigated the separation values that these permit, though. Physically skewing the tubs may also allow for replication although I'm yet to find an explicit example.

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x = 3, y = 54, rule = B3aeiy4eqrty5cijn6ck7c8/S2-i3acein4cejkq5ir6cei7c8
My main question is: is it possible to find any simpler examples of adjustable-speed replicators? It certainly seems like a much more difficult task than finding adjustable spaceships, given that's been done plenty of times so far. The piston-like design of the original adjustable spaceships rule will probably be quite hard to capture, especially if we're to be able to copy the period information to an infinite number of successive copies and have them all cleanly annhialate. So I can't be too sure. But my main target right now is for an easy to describe and extend family of replicators to be found, such as, for example, all speeds c/2n below a given speed. Anyone up for this challenge?