Dodecalplex, Hopf Meridian w South & Infinity

The normal Hopf fibration of an hypersphere consists in an infinite family of circles (fibers) where each fiber is distinct, any two fibers are interlinked and all fibers generate the entire hypersphere.
When a stereographic projection is applied, the hypersphere maps with the infinite Euclidian 3D volume.

A discrete Hopf fibration can be found with regular polytopes (equivalent of platonic solids, but in 4D).
In particular, the Hopf fibration in the 120-cells (Dodecaplex) consists of 12 rings of 10 dodecahedra each : https://en.wikipedia.org/wiki/120-cell

The cell-centered stereographic projection of the dodecahedron offers a nice visualization of this fibration.
We can see 4 kinds of fibers :
- the south pole fiber (a vertical axis circle of 10 dodecahedra and a minimal radius)
- the north pole fiber (an horizontal axis circle with an infinite radius)
- 5 south hemisphere fibers (adjacent to the south pole fiber)
- 5 north hemisphere fibers (adjacent to the north pole fiber)

In this discrete fibration, the infinite cell is the cell opposite to the centered cell. It is a dodecahedron with 12 pentagonal faces oriented inward rather than outward (the center of the cell is at infinity). The north pole fiber is a ring with both the center cell (normal dodecahedron) and the dodecahedron at infinity.

The 120 cells are organized in 9 layers (L1 being the cell centered dodecahedron and L9 the cell at infinity).
One nice way to visualize these 9 layers is by dividing the polytope though the meridian.
In an hypersphere, the meridian cuts the object in two exact halves.
In this discrete fibration, and since the symmetry is order 5, it is not possible to divide the 120 cells though the meridian without cutting cells.

A model is proposed with 2 asymmetrical halves following as closely as possible the meridian. Only the cell at infinity is cut in half in this visualization.

The 120 cells (or dodecaplex) is a very nice object and the base of a very nice puzzle you can find on Maths Gears (https://mathsgear.co.uk/products/dodecaplex-puzzle) with The Ring and The Pulsar being shown in the illustration.

6 versions are presented :
- 1 ring of each kind, without the dodecahedron at infinity (exist in color)
- 1 ring of each kind, with a part of the dodecahedron at infinity (exist in color)
- meridian with the southern pole ring
- meridian with the southern pole ring, and part of the dodecahedron at infinity
- meridian with the northern pole ring, without the dodecahedron at infinity
- meridian with the northern pole ring, and part of the dodecahedron at infinity