Synergetics is actually easy. You know that you can fit six circles around a center circle. If you doubt it, try with seven coins of the same denomination.
Turns out with spheres, you can add six more spheres with three above and three below the center sphere. This is Fuller's Vector Equilibrium. The constraining bands and the radiating spokes are equal in number (12).
http://en.orion-metaphysics.com/wp-c...ctahedron3.gif
This is an Archmedian semi-regular, but it can 'jitterbug' into two of the Platonic solids. That was one of Bucky's favorite demonstrations.
https://i.pinimg.com/originals/7e/aa...837fff71b3.gif
The icosahedron is an intermediate phase when the squares turn into two triangles; then on to the octahedron.