Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space. Issue 3 (13th May 2021)

• Record Type: Journal Article
• Authors: Zamboj, Michal
• Is Part Of: Journal of computational design and engineering. Volume 8:Issue 3(2021)
• Page Start: 836
• Page End: 854
• Publication Date: 2021-05-13
• Subjects: Hopf fibration -- Hopf tori -- four-dimensional visualization -- stereographic projection -- synthetic construction; Engineering -- Data processing -- Periodicals; Computer-aided design -- Periodicals; Computer-aided design; Engineering -- Data processing; Electronic journals; Electronic journals; Periodicals; 620.0042
• DOI: 10.1093/jcde/qwab018
• Languages: English
• ISSNs: 2288-4300

Abstract

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. Graphical Abstract: