File:Riemann Sphere.gif

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File:Riemann Sphere.gif - Wikipedia

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Summary

DescriptionRiemann Sphere.gif	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	24 June 2019
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

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	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. CC0falsefalse

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	08:04, 26 June 2019		360 × 286 (710 KB)	Berto	User created page with UploadWizard

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GIF file comment	Created with the Wolfram Language : www.wolfram.com

Summary

Mathematica 11.0 code

Licensing

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set of complex numbers

mapping

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some value

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copyrighted, dedicated to the public domain by copyright holder

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Creative Commons CC0 License

inception

24 June 2019

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image/gif

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