I just stumbled across a way to build a 7-way radial symmetry in LEGO.
A technique for building 7-way radial symmetry in LEGO. The blue 2×2 plates are connected to each other using plate hinges in a ring, and when I built that I noticed the ring looked like the right diameter to go around a 4×4 round piece. The hinges actually protrude a little into the center, but by sandwiching the two 4×4 rounds together and filling the space inside with a round 2×2 plate and seven (leaving one stud open) 1×1 round plates, you can make it all fit nicely. And since the 4×4 rounds and 2×2 round plate have axle holes, it can easily be connected to any Technic mechanism.
If you stick an axle in the middle it twirls like a top – it’s very well balanced.
LDraw code after the break: