The (diminished) 120-Cell and it’s relationships to the 5-Cell (A4), the 600-Cell (H4), the two 24-Cells (D4), the Dual Snub 24-Cell, and of course E8!

The paper being referenced in this analysis is here.

I’ve added this PDF or this Mathematica Notebook (.nb) which is having a bit of fun visualizing various (3D) orbits of diminished 120-cell convex hulls. The #5 subset of 408 vertices (diminishing 192) is completely internal to the normal 3D projection of the 120-cell to the chamfered dodecahedron. The #3 has 12 sets of 2 (out of 3) dual snub 24-cell kite cells (that is 24 out of the 96 total cells).