Assume the inner one remains a hyperboloid ruling (made out of wires or something the strings won't affect much). What shapes do the outer ones make? I don't think it's quite a ruling any more if the strings bend, but I haven't simulated it yet.
@Sabetta_ two concentric hyperboloids, made from string and wood rings: both of different radii when untwisted, coaxial, same distance between rings.
twist the inner one a little. twist the outer one until its strings touch the inner one's strings.
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(this idea has been lurking for a few weeks, and I just saw https://youtu.be/0e5vzTsLh2s )
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