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- Title
Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space.
- Authors
Zamboj, Michal
- Abstract
The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in 4-space using computer graphics based on their analytical representations has become popular in recent decades. For purely synthetic constructions, we apply the recently introduced method of visualization of 4-space by its double orthogonal projection onto two mutually perpendicular 3-spaces to investigate the Hopf fibration as a four-dimensional relation without analogy in lower dimensions. In this paper, the method of double orthogonal projection is used for a direct synthetic construction of the fibers of a 3-sphere from the corresponding points on a 2-sphere. The fibers of great circles on the 2-sphere create nested tori visualized in a stereographic projection onto the modeling 3-space. The step-by-step construction is supplemented by dynamic three-dimensional models showing simultaneously the 3-sphere, 2-sphere, and stereographic images of the fibers and mutual interrelations. Each step of the synthetic construction is supported by its analytical representation to highlight connections between the two interpretations.
- Subjects
H-spaces; EUCLIDEAN distance; COMPUTER graphics; ORTHOGRAPHIC projection; TOPOLOGY
- Publication
Journal of Computational Design & Engineering, 2021, Vol 8, Issue 3, p836
- ISSN
2288-4300
- Publication type
Article
- DOI
10.1093/jcde/qwab018