The 24-Cell and its Snub

Here are two Zometool models of the 24-cell. The first is an all-yelow rhombic dodecahedron with some added yellow and blue struts, but uses no red struts. The second uses blue, yellow, and red struts. In the first model, twelve of the octahedra appear completely flattened as 2-dimensional rhombii, where the ratio of the diagonals of each rhombus is the square-root of two. In the second model, all of the images of the octahedra under the canonical Zomic projection possess positive 3D volume, so it is perhaps easier to see how the octahedra are attached along their triangular faces.

The 24-cell has a "snub" version, analogous to the snub cube or snub dodecahedron. This has 24 regular icosahedra and 120 regular tetrahedra. The tetrahedra comprise two orbits under the action of the symmetry group. One of these orbits has 24 tetrahedra and these coincide with the octahedra in the dual of the 24-cell. The other orbit has 96 tetrahedra. One should imagine 24 assemblies of five tetrahedra, where one tetrahedra is attached to four neighbors.

With these models, one may see that the 24-cell can be obtained from the 600-cell by removing select vertices and incident edges. For that matter, one sees that the vertex figure of the 600-cell is a regular icosahedron.