Six axis racks

In this symmetric mechanism, moving any of the twelve sticks into or out of the core moves all of the others. The force is transferred around the core by gears meshing with racks on the sticks.

To build this you will need twelve copies of the stick part, twelve copies of the gear part, twelve copies of the cap part, six copies of the core part, six M3 nuts, and six 45mm long M3 bolts. The core part requires a small patch of support in the center, but otherwise all parts should print without support.

First, assemble the six core parts, in the same way that the star burr puzzle is assembled. (Thanks to Saul Schleimer for the idea to use this design for the core). Next bolt the gears onto the core. Each bolt goes through (in order) a cap part, a gear, two core parts, a gear, and a cap part. Finally, thread the sticks through the core. 

I found this last step very fiddly to do, particularly if you want the sticks to be lined up the same distance into the core as each other. I frequently loosened bolts and lifted gears to move the racks into better positions. The racks also have two different ways that they can be inserted. Check the photos carefully to see the orientation that allows the distances of the sticks into the core to be the same.