As it turns out, eight icosahedra form this rhombic ring, by augmentation:
Measured from the centers of these icosahedra, the long and short diagonal of this rhombus are in a (√2):1 ratio. How do I know this? Because that’s the only rhombus which can made this polyhedron, a rhombic dodecahedron, dual to the cuboctahedron.
This rhombic dodecahedral cluster of icosahedra could be extended to fill space, since the rhombic dodecahedron itself has this property, an unusual property for polyhedra. Whether space-filling or not, the number of icosahedron per rhombic-dodecahedron edge could be increased to 5, 7, 9, or any greater odd number. Why would even numbers not work? This is a consequence of the fact that opposite faces of an icosahedron are inverted, relative to each other; a pair of icosahedra (or more than one pair, producing odd numbers > 1 when added to the vertex-icosahedron) must be attached to the one at a rhombic-dodecahedron-vertex to make these two inversions bring the triangular face back around to its original orientation, via an even number of half-rotations, without which this consruction of these icosahedral rhombi cannot happen.
Here’s another view of this rhombic dodecahedron, in “rainbow color” mode:
All images above were produced using Stella 4d, software which may be tried for free right here.
Ten rhombic triacontahedra fit perfectly into a decagonal ring. It’s not a “near-miss” — the fit is exact.
I made this with Stella 4d, software you can try for free, or purchase, at http://www.software3d.com/Stella.php.
This solid, also known as the great rhombicuboctahedron, and the truncated icosidodecahedron, can be used to build many other things. In addition to the elongated ring of eight above, for example, there’s this octagonal prism.
Remember the elongated ring at the top of this post? This pic, directly above, is of a ring of four of those rings.
And, yes, that’s a (non-great) rhombcuboctahedron made of great rhombcuboctahedra. Here it is again, with a different color-scheme.
For the last of these constructions, eight more great rhombcuboctahedra are added to the figure in the two posts above, which is also returned to its original color-configuration. These eight new polyhedra have positions which correspond to the corners of a cube.
Manipulating polyhedra in this manner is easy with Stella 4d, the program I used to do all of this. You may buy it, and/or try a free trial version first, at www.software3d.com/Stella.php.
Software credit: see http://www.software3d.com/stella.php.
I made this .gif with Stella 4d, software available as a free trial download at http://www.software3d.com/stella.php.