Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 61 icosahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.
I made this using Stella 4d, software you can try for yourself right here.
These two polyhedra are the icosahedron (left), and the great icosahedron (right).
Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.
I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.
The polyhedra at the vertices are rhombic triacontahedra, and the yellow edges are elongated rhombic prisms. This was made using Stella 4d, software you may try for free at this website.
This was made using Stella 4d, software available at http://www.software3d.com/Stella.php.
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.
I used Stella 4d: Polyhedron Navigator to make this. You can find this program at http://www.software3d.com/Stella.php.