Hopf Fibration, North, Orthographic projection

The Hopf fibration is related to the hypersphere (the 4D equivalent of the sphere).
The usual sphere is a 2D manifold (surface) and the hypersphere is a 3D manifold (volume).

It cannot be represented directly because it is the boundary of a 4D object (a 4D ball).
The Hopf fibration uses a nice property of the hypersphere to visualize it with a set of circles (fibers).
These circles never overlap and are all interlinked 2 by 2.

A projection into our 3D world is required to visualize it.
It can be done with an orthographic projection (all circles have the same radius) ot with a stereographic projection.

For the projection, a point of view (point + axis) is required which defines two circles as a north pole (our eye) and a south pole (the furthest fiber from the north pole).

In this design, the stereographic projection is done by keeping the circle size constant at the equator.
At the south pole, the circle size is reduced.
At the north pole, the circle size tends to infinity and is represented by a straight line (infinity radius).

The meridian is the set of fibers that goes directly from the south pole to the north pole.
In our projections, it is a surface.
When this surface is rotated around the south pole - north pole axis, it create the full set of fibers.
In the orthographic projection, it gives a solid horn torus.
In the stereographic projection, it gives a solid 3D euclidian space (fibers fill the entire volume of the 3D universe).

The design consists of equally spaced fibers along the meridian (all circles).
Some additional lines are added to underline the meridian surface geometry.
Each of the meridian fiber generates a torus (surface) when rotated along the south-north axis, the fiber (circle) being one of the Villarceau circle of the torus.

The equator is the Clifford torus (surface) midway in the meridian with 45° Villarceau circles.

The meridian and the equator both divides the full volume in 2 equivalent solid torus (2 Clifford torus).
They are distorted in our projection, the southern solid torus being the less distorted (solid torus between 2 surfaces, one being a no "thickness" singularity / cricle).
In the stereographic projection : the northen solid torus is between 2 surface, one being a singularity at infinity.
When divided by the merdian, the two torus meet at the 3D printed surface. Their other surfaces are singularities at infinity.
A nice animation enables to better understand this kind of torus inversion (https://en.wikipedia.org/wiki/Torus#/media/File:Clifford-torus.gif)

5 versions are presented : 
- Orthographic projection, full meridian : all circles have radius 1 cm
- Orthographic projection, southern meridian : the circles form an hopf band (a mobius strip with 2 twists)
- Orthographic projection, northern meridian
- Stereographic projection : southern meridian + northern meridian up to latitude 50° (full circles). After latitude 50° north, only a portion of the meridian is represented (partial circles). The cut is done in in order to underline the symmetry of this surface.
- Stereographic projection : same but with a square frame