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

The result's identifiers

  • 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>

Alternative languages

  • 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

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • OECD FORD branch

    50301 - Education, general; including training, pedagogy, didactics [and education systems]

Result continuities

  • Project

  • Continuities

    I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Others

  • Publication year

    2021

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Journal of Computational Design and Engineering [online]

  • ISSN

    2288-5048

  • e-ISSN

  • Volume of the periodical

    8

  • Issue of the periodical within the volume

    3

  • Country of publishing house

    NL - THE KINGDOM OF THE NETHERLANDS

  • Number of pages

    19

  • Pages from-to

    836-854

  • UT code for WoS article

    000657566100003

  • EID of the result in the Scopus database

    2-s2.0-85107796474