> Non-spherical geodesic domes -- dual to figures like the attached --
> where the pentagonal hubs are *not* arranged as the vertices of an
> icosahedron.  The "parts" I mentioned are the hubs, the "nodes" are the
> tips of five or six radiating rods; each tip seeks a corresponding tip.
>
> (In general the nodes won't exactly match, and even if they did there
> would be an angle between rods; cubic splines will cover both flaws.)
>
>> and what you would like help with.
>
> I want to know how to generate, for each part, a rotation-translation
> matrix that brings its nodes nearer to their mates.

In that case I recommend you follow the 2nd solution in my original reply.

> (Scare-quotes because I don't want to give the body momentum, linear or
> angular; I want it to jump to a new position and wait quietly for the
> next cycle.)
> Torque is a vector, and vectors can be added and rescaled.
> But how do I turn the resulting torque vector into a rotation matrix?

The "short-cut" you are trying to find would only work if the resultant 
torque on the body were constant. But because the forces on the nodes 
are causing the body to rotate, the resultant torque will be 
continuously changing. The only way to solve this is to integrate the 
torque to get angular momentum, then integrate again to get the rotation.

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