The icosidodecahedron has a long and interesting stellation-series, and you can see the whole thing using Stella 4d, the program I used to make the rotating .gifs here. Rather than keep the scale the same in each frame, I set the program to make the polyhedron as large as possible, while still fitting in the image-box. This creates the illusion that the polyhedra below are “breathing.”

The polyhedron above is the 20th stellation of the icosidodecahedron — the one that appeared as the sole image in the last post here, but with completely different colors. The next one shown is the 31st stellation.

The 55th stellation is immediately above, while the next one is the 69th.

The 84th stellation is immediately above, while the next one is the 89th.

The 106th stellation is immediately above, while the next one is the the 110th.

The 135th stellation is immediately above, while the next one, which is chiral, is the 157th.