I was watching some 3Blue1Brown videos and came across one about the Fourier transform which illustrated by drawing arbitrary pictures as complex functions with, I think, a constant pen velocity. The complex Fourier transform amounts to representing the function as a sum of rotating phasors, so the dude just drew the phasors; their magnitudes and initial phase wholly determine the image and the pen speed.
It occurred to me that some kind of mechanism vaguely like this could be used for cutting arbitrary toolpaths, like rosette "machine turning". To keep it balanced, you'd want each phasor to be not a single arm rotating around one end, but a bar rotating around its center, with a big enough counterweight at the other end to counterbalance the whole assembly of succeeding phasors at its business end. This quickly gets into exponential growth so you don't want to have too many levels of phasor.
Probably, though, in a physical machine, you will want to vary not the radius of each rotation but its speed, since that's what you control more directly. This poses an interesting optimization problem of how to trace some desired toolpath using such a balanced kinetic chain with fixed radii by setting the rotations to specific speeds.
You can get twice as many degrees of freedom by moving the workpiece as well as the tool, but this involves carefully adjusting the counterweights to match the workpiece's mass.