Hyperboloid!

There were two main challenges in the prototype:

  1. Figuring how to evenly distribute 6 points around a central point
  2. Devising a central axis that can increase and decrease in length as the model twists
Prototype extended Prototype collapsed Prototype from above
Prototype

(Gotta love universal joints.)

The final model uses 32L axles and 11 x 11 gear racks (major kudos to the author of this article about its geometry):

Model extended Model collapsed Model from above
Hyperboloid

Video of the final model: