This project is an exercise in tetrahedral symmetry, and as such is not dissimilar to previous projects like this and this. The principle is always the same: find a polyhedron having at least three faces in the same planes as the faces of a tetrahedron, then attach the polyhedron to 19 objects just like it by means of those faces. You can see in the photograph that I used the same gray color for all of the edges of the shared triangles.
The rest of the coloring scheme, however, comes to you courtesy of LMAbacus. You see, a few months ago, he not only made the astute observation that it would be possible to spread a rainbow gradient across the whole of a dodecahedron by means of its vertices, but he also took the time to put together a wonderful chart demonstrating how to accomplish the deed. Bolstered by this, I finally decided to buy a dozen packs of the most diverse origami paper I could find. Then, for the next few months, the coloring guide served as a roadmap dictating which colors to fold and when. Though I sometimes failed to find a good match for the colors that I needed, I still think that the end result looks fantastic.
And, yes, it has indeed been months. I finished folding this on September 25th, and I think I started at around July 5th, so this has taken around 81 days to fold. That is far longer, I think, than the time dictated by this project's complexity. The only reason it has taken so long is because I was lazy; I spent several weeks not even touching the project simply because I didn't feel like working on it! Whenever I actually bothered to spend a few solid hours on the project, a single rhombicuboctahedron would generally take me about three hours to fold and assemble.
There are only 870 units in this object. I did not count the paper, so how did I come up with that number?