Download Blender 2.9 (or later) for free here: https://www.blender.org/download/
The Hopf Fibration is the fibre bundle created by projecting the four-dimenstional sphere S3 onto the three-dimentional sphere S2, every point on S2 has a corresponding fibre in three-dimensional space R3 which is a perfect two-dimensional sphere (or circle) S1. This structure is written as S1 -> S3 -> S2, which can be read as; "The fibre-space S1 is embedded in the full-space S3 which is projected to the base-space S2".
This fibre bundle is space-filling in R3, every fibre links with every other fibre and no two fibres ever intersect. Lines of constant elevation (ε) on S2 are seen as tori in the fibre-space, lines of constant azimuth (α) appear has Hopf bands. There are two edge cases at the poles of S2; at the south pole (α = 0) the fibres converge to a circle lying flat in the X-Y plane, at the north pole (α = π) the fibre is a circle through infinity (here visualised as the straight line lying on the Z-axis).
The file fibration.py is a visualisation tool which uses the Python Blender API.
This includes the function makeFibre(elevation,azimuth) which creates the fibre corresponding to the point on the sphere S2 at the coordinate (ε,α).
Fibres are visualised as tori in order to be visible, the cross-sectional diameter can be adjusted with the optional argument sectionRad. Also included is a function to convert (ε,α) into RGB colours for visualisation.
Figure 1 Fibres at 100 equally spaced azimuths and 8 equally spaced elevations. This results in 6 nested tori and the two edge cases.
Figure 2 As in Figure 1 but fewer fibres are rendered showing a cross-section.
Figure 3 25 fibres all at ε = π/2. Two fibres are shown in pink to highlight how the fibres link but do not intersect. Edge cases also shown in blue and red.
Figure 4 An earlier colouring scheme, I have still not finalised this, there is a lot to play around with.
Simply make a new Blender file or download the provided example(fibration.blend), create a Text Editor pane, open the fibration.py file and use the keyboard shortcut Alt+P to run the script.
This will run an example visualisation which generates fibres constructing six tori and the two edge cases at ε = {0,pi/2} (a flat cricle and a circle through infinity respectively).
- Improve the automatic colouring of fibres
- Add a visualisation of S2 showing points corresponding to rendered fibres of the same colour
- Add functionality for easy animation creation