File:Bednorz Sphere Eversion of Earth.webm
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DescriptionBednorz Sphere Eversion of Earth.webm |
English: This video shows a sphere eversion using the method described in the following article: Adam Bednorz, Witold Bednorz: Analytic sphere eversion using ruled surfaces, Differential Geometry and its Applications 64:59-79, 2019.
An eversion turns the sphere inside-out in a smooth manner. Creases, pinch points, holes, etc. may not occur during the eversion. However, the sphere may intersect itself during the eversion. In mathematical terms, the eversion is a regular homotopy between the sphere and the sphere point reflected at its center. Instead of the usual approach of coloring the two sides of the sphere with different colors, e.g., as in File:Evshort2.webm, the sphere's inside is painted with an image of the earth by night and the sphere's outside is painted with an image of earth by day. This shows the deformation of the sphere that happens during the eversion more clearly. Furthermore, this shows that after the eversion has completed, there is actually a point reflection that reverses the orientation of the sphere. Note that the continents on the night side are upside-down and mirror-reversed. Note that the video can be played in a loop using a suitable video player.Deutsch: Dieses Video zeigt eine Umstülpung der Sphäre mit der Methode, die in folgendem Artikel beschrieben wird: Adam Bednorz, Witold Bednorz: Analytic sphere eversion using ruled surfaces, Differential Geometry and its Applications 64:59-79, 2019.
Die Sphäre wird von innen nach außen umgestülpt, wobei die deformierte Sphäre zu jedem Zeitpunkt mathematisch glatt ist. Falten, Zwickpunkte, Löcher oder ähnliches dürfen bei der Umstülpung nicht auftreten. Die Sphäre darf sich jedoch während der Umstülpung selbst durchdringen. Mathematisch betrachtet ist die Umstülpung eine reguläre Homotopie zwischen der Sphäre und ihrer Punktspiegelung an ihrem Mittelpunkt. Statt des üblichen Ansatzes, die Innen- und Außenseite der Sphäre mit zwei verschiedenen Farben zu färben, wie z. B. bei File:Evshort2.webm, wird hier die Innenseite der Sphäre mit einem Bild der Erde bei Nacht und die Außenseite mit einem Bild der Erde bei Tag eingefärbt. Dies zeigt die Deformation der Sphäre während der Umstülpung deutlicher. Außerdem zeigt dies, dass tatsächlich eine Punktspiegelung der Sphäre vorliegt, nachdem die Umstülpung vollendet wurde. Dies kann man daran erkennen, dass die Kontinente auf der Nachtseite auf dem Kopf stehen und spiegelverkehrt sind. Das Video kann mit einem geeigneten Videoplayer in einer Schleife abgespielt werden. |
Date | |
Source | Own work |
Author | Carsten Steger |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 15:59, 9 January 2022 | 1 min 25 s, 1,080 × 1,080 (24.96 MB) | Carsten Steger (talk | contribs) | Uploaded own work with UploadWizard |
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Short title | Sphere eversion of the earth |
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Copyright holder | © 2021 Carsten Steger. This work is licensed under the Creative Commons Attribution ShareAlike 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by-sa/4.0/. |
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