For the 4D polytope 5-cell (simplex), this is the projection to 3D with the maximal volume. This is an edge-first projection.

5-cell max volume projection

For the 4D polytope 8-cell (tesseract), this is the projection to 3D with the maximal volume. This is a vertex-first projection.

8-cell max volume projection

For the 4D polytope 16-cell (cross polytope), this is the projection to 3D with the maximal volume. This is a vertex-first projection.

16-cell max volume projection

For the 4D polytope 24-cell, this is the projection to 3D with the maximal volume. It appears random. However, one of the 2D projection of this 3D shape, with a 6-fold symmetry, is the max area 2D shadow of the 24-cell.

24-cell max volume projection

For the 4D polytope 600-cell, this is the projection to 3D with the maximal volume. Just like the 24-cell, it appears random. However, one of the 2D projection of this 3D shape, with a 10-fold symmetry, is the max area 2D shadow of the 600-cell.

600-cell max volume projection

For the 4D polytope 120-cell, this is the projection to 3D with the maximal volume. It appears random and I don’t really see any connection to other known projections.

120-cell max volume projection

Thank you, Scott V, David Hall, for the discussion and some of the models here.