It is well known that six carbon atoms can form a ring called cyclohexane. Since the angle between bonds is $\cos^{-1}\left(\frac{-1}{3}\right)\approx 109^\circ$, the ring is not a planar hexagon. There is a flexible configuration called the boat and a rigid one called the chair.
I am interested in the $3D$ geometry. The chemistry is modeled by rods attached at nodes so that the angle at which two rods meet is fixed but dihedral angles are free to change.
Question: In such a structure, where all the rods are equal and all the angles are equal, for which angles does there exist a flexible structure? Furthermore, classify all such structures, flexible or rigid.
I know that for $120^\circ$, the planar hexagon, there are no flexible structures.
Conversations with several chemists have yielded no information.