Visualizing objects of four-dimensional space: From flatland to the Hopf fibration

Kód výsledku v IS VaVaI

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Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Visualizing objects of four-dimensional space: From flatland to the Hopf fibration

Popis výsledku v původním jazyce

One of the fundamental questions of a three-dimensional geometer is how to imagine a four-dimensional object. And yet, he draws pictures of three-dimensional objects in the two-dimensional paper. Moreover, would a two-dimensional geometer understand our sketches? Based on analogies, we give an overview of methods of examination of four-dimensional objects. We emphasize visualization as the main element of perception of four-dimensional space. For this purpose, we describe the double orthogonal projection of the four-dimensional space onto two mutually perpendicular three-dimensional spaces as a generalization of the classical Monge&apos;s projection. In such a projection, we construct a four-dimensional playground for convenient synthetic creation of four-dimensional objects. All our constructions are easily accessible with the interactive 3D modeling software GeoGebra. Furthermore, we apply the method of projection to an intuitive investigation of various four-dimensional mathematical phenomena - polytopes, four-dimensional quadrics, three-sphere and its stereographic projection, complex plane, and the Hopf fibration.

Název v anglickém jazyce

Visualizing objects of four-dimensional space: From flatland to the Hopf fibration

Popis výsledku anglicky

One of the fundamental questions of a three-dimensional geometer is how to imagine a four-dimensional object. And yet, he draws pictures of three-dimensional objects in the two-dimensional paper. Moreover, would a two-dimensional geometer understand our sketches? Based on analogies, we give an overview of methods of examination of four-dimensional objects. We emphasize visualization as the main element of perception of four-dimensional space. For this purpose, we describe the double orthogonal projection of the four-dimensional space onto two mutually perpendicular three-dimensional spaces as a generalization of the classical Monge&apos;s projection. In such a projection, we construct a four-dimensional playground for convenient synthetic creation of four-dimensional objects. All our constructions are easily accessible with the interactive 3D modeling software GeoGebra. Furthermore, we apply the method of projection to an intuitive investigation of various four-dimensional mathematical phenomena - polytopes, four-dimensional quadrics, three-sphere and its stereographic projection, complex plane, and the Hopf fibration.

Druh

D - Stať ve sborníku

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název statě ve sborníku

Proceedings of the 19th Conference on Applied Mathematics, APLIMAT 2020

ISBN

978-80-227-4983-1

ISSN

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e-ISSN

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Počet stran výsledku

25

Strana od-do

1140-1164

Název nakladatele

Slovak University of Technology in Bratislava

Místo vydání

Bratislava

Místo konání akce

Bratislava; Slovakia

Datum konání akce

4. 2. 2020

Typ akce podle státní příslušnosti

EUR - Evropská akce

Kód UT WoS článku

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