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Riemann_Sphere.gif (360 × 286 pixels, file size: 710 KB, MIME type: image/gif, looped, 126 frames)

Summary

Description
English: A complex number c can be mapped on the surface of a sphere, where the south pole corresponds to c=0+i0, the equator to |c|=1, and the north pole to complex numbers with infinite modulus (their phase doesn't matter).
Date
Source https://twitter.com/j_bertolotti/status/1143077137861230592
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

riemannsphere[c_] := {(2 Re[c])/(1 + Re[c]^2 + Im[c]^2), (2 Im[c])/(1 + Re[c]^2 + Im[c]^2), (Re[c]^2 + Im[c]^2 - 1)/(1 + Re[c]^2 + Im[c]^2)};
p1 = Table[
   c =3 Sin[2 j] + I 2 Sin[1 j];
   Graphics3D[{Text[Style["Re", Bold, Large], {3.4, 0, 0}], Text[Style["Im", Bold, Large], {0, 3.6, 0}], Thick, Arrow[{{-3.1, 0, 0}, {3, 0, 0}}], Arrow[{{0, -3.1, 0}, {0, 3, 0}}], Line[{{0, 0, 1}, {Re[c], Im[c], 0}}], Line[{{0, 0, 1}, riemannsphere[c]}], PointSize[0.03], Blue, Point[riemannsphere[c] ], Black, Point[{Re[c], Im[c], 0}], Opacity[0.5], White, Sphere[], Polygon[{{3, 3, 0}, {3, -3, 0}, {-3, -3, 0}, {-3, 3, 0}}]}, Boxed -> False, Lighting -> "Neutral", ViewPoint -> {3, 3, 3}, PlotLabel -> Style[StringForm["c\!\(\*SubscriptBox[\(=\), \(``\)]\)", Round[c, 0.01], j], Medium, FontFamily -> "DejaVu Serif"], LabelStyle -> {Black, Bold} ]
   , {j, 0, 2 \[Pi], 0.05}];
ListAnimate[p1, 15]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

Matting of the complex plane on the Riemann sphere

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depicts

24 June 2019

image/gif

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Date/TimeThumbnailDimensionsUserComment
current08:04, 26 June 2019Thumbnail for version as of 08:04, 26 June 2019360 × 286 (710 KB)BertoUser created page with UploadWizard

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