The Direct graphics and movies show the last two, the expansion stages. These are intuitive visions. That is, after "seeing", imagining, and visualizing the forms, it takes time to identify the component parts and verify relationships. Faces of the 3D block and inner tetrahedrons appear expanded in the haptihedron, at the start of this movie. However, what accounts for the 12 square faces between?
The EIE has 20 triangles (icosahedron) and 12 pentagons (dodecahedron). In transition to the EIE, the 6 cube faces divide to become 12 triangles. With the 8 triangles from the tetrahedrons, they form the 20 faces of an icosahedron. Yet what accounts for the 12 pentagonal faces of the EIE? The movie shows how the 12 square faces of the haptihedron evolve into the 12 pentagonal faces of the EIE. Each of the squares needs another side to become a 5-sided pentagon. It looks as though the original cube face is the source.
Look at it closely. Each original cube face has 4 sides. Focus on the top blue square (in perspective, a diamond) and watch as it transforms. In the EIE, the original square now has 6 sides, three for each triangle. All sides border on the pentagons. As the two sides of the square divide, they provide extra sides for each neighboring square, to create pentagons in EIE.
For more ways to study the geometry, see the evolving brain cell geometry page.