Applet below shows central cross sections of 64-cells 4-dimensional antiprism from lower to upper base (for definition and basic description see a page of 'Literka': 4-dimensional antiprisms.)
To get basic ideas about 3-dimensional antiprisms see 3-dimensional antiprisms.
Pictures of the applet below are interesting for another reason. They present polyhedrons morphing beginning from a regular icosahedron transforming into 62-faces semi-regular polyhedron, then a regular dodecahedron.