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.

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## About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet.
The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.

Nice one Robert.

Leslie

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Thank you! =)

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