The image of Saturn was taken by NASA, and I put it on the faces of a rhombic dodecahedron, and created this image, with a program called Stella 4d. You can try this program for free at http://www.software3d.com/Stella.php.
The truncated octahedron is well-known as the only Archimedean solid which can fill space, by itself, without leaving any gaps. The cluster below shows this, and has the overall shape of a rhombic dodecahedron.
It’s easier to see the rhombic dodecahedral shape of this cluster when looking at its convex hull:
Both images here were made using Stella 4d, which you can try for free right here.
Programs used to make this included Geometer’s Sketchpad, MS-Paint, and Stella 4d: Polyhedron Navigator, which was used to assemble everything else into what you see here. You may try Stella for free at http://www.software3d.com/Stella.php.
I used Stella 4d to make this polyhedral compound, and this program may be tried for free at this website.
Created using Stella 4d, a program available at this website: http://www.software3d.com/Stella.php.
I made this using Stella 4d, software you can find at http://www.software3d.com/Stella.php. I also make a second version, with larger spheres and cylinders for the vertices and edges:
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.
