Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11410%2F21%3A10428598" target="_blank" >RIV/00216208:11410/21:10428598 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=-oHAE259ZM" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=-oHAE259ZM</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1093/jcde/qwab018" target="_blank" >10.1093/jcde/qwab018</a>

Result language
angličtina
Original language name
Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
50301 - Education, general; including training, pedagogy, didactics [and education systems]

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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