From pole to pole
Introduction to Hopf Fibration Mappings
Welcome to the intriguing world of Hopf Fibration Mappings. Here, we use points on a regular three-dimensional sphere to venture into the heart of a four-dimensional sphere, known as S3. If you haven´t read the general post on Hopf Fibration here is a small summary:
Transformation Process
This mapping journey relies on transformation. By focusing on strategically placed reference curves on the three-dimensional sphere, we can avoid representing the entire surface.
Rotating circle
Role of ‘Fibers’
At the core of this process are ‘fibers,’ which initially resemble circles. However, their shape changes based on their proximity to the ‘projection center.’ The closer the fibers are to this center, the more they retain their circular shape. As they drift away, they begin to distort, even stretching into lines at extreme distances.
Circle horizontal
Visual Projection
Our videos offer a captivating view of this transformation. By projecting drawings onto a 3D map, and then onto the Hopf Fibration, we connect each colored dot on the sphere to a corresponding curve in space. As the dots shift relative to the projection center, their representations in the projection transform.
Circle closing
Intersection of Villarceau Circles
It’s important to note that all these fibers technically qualify as Villarceau Circles. Consequently, any two points on the spherical map will always intersect, forming interlocking circles within the projection.
Arc rotating
Implications and Future Exploration
The potential applications of this discovery are vast, something we’ll delve into in future posts. Hopf Fibration Mappings, thus, serve as a critical tool for understanding higher dimensions.
Arcs moving
Conclusion
In conclusion, Hopf Fibration Mappings offer a unique perspective into the complexities of higher dimensions. Join us as we continue to simplify and explore the world of higher-dimensional mappings in upcoming posts.