Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2020
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914449219125248 |
|---|---|
| author | Zamboj, Michal |
| author_facet | Zamboj, Michal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_09236 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space Zamboj, Michal History and Overview Differential Geometry 51N05 (Primary) 55R10 65D17 (Secondary) 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. |
| title | Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space |
| topic | History and Overview Differential Geometry 51N05 (Primary) 55R10 65D17 (Secondary) |
| url | https://arxiv.org/abs/2003.09236 |