File:Metric vs pi.gif
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Metric_vs_pi.gif (480 × 480 pixels, file size: 2.03 MB, MIME type: image/gif, looped, 252 frames, 25 s)
Captions
Captions
The value of pi depends on the metric.
Summary
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| Description |
English: π is the ratio between the circumference and the diameter of a circle, but what you call a "circle" depends on how you calculate distances. If instead of the usual Euclidean norm, we use one of the p-norms, we get different values for π.
(A p-norm takes the usual Euclidean norm ||x⃗||²=(x₁²+x₂²) and generalizes it to ||x⃗||ᵖ=(x₁ᵖ+x₂ᵖ), where p is a real number ≥1). Reference: https://jstor.org/stable/2687579 |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1503407809085460480 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 13.0 code
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frames = Table[
tmp = NIntegrate[2/p (u^(1 - p) + (1 - u)^(1 - p))^(1/p), {u, 0, 1}];
ContourPlot[Evaluate[Norm[{x, y}, p]] == 1, {x, -1.3, 1.3}, {y, -1.3, 1.3}, Frame -> False, ContourStyle -> Directive[Thick, Black], Epilog -> {Text[Style[StringForm["\[DoubleVerticalBar]!\(\*OverscriptBox[\(x\), \(\[Rule]\)]\\)\[DoubleVerticalBar]=\!\(\*RadicalBox[\(\*SuperscriptBox[\SubscriptBox[\(x\), \(1\)], \(``\)] + \\*SuperscriptBox[SubscriptBox[\(x\), \(2\)], \(``\)]\), \(``\)]\) \
\[Rule] \[Pi]=``", NumberForm[p, {3, 2}], NumberForm[p, {3, 2}], NumberForm[p, {3, 2}], NumberForm[tmp, {6, 5}]], Black, Bold, FontSize -> 18], {0, 1.125}]}
]
, {p, 1, 7, 0.05}];
ListAnimate[frames]
Licensing
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I, the copyright holder of this work, hereby publish it under the following license:
 
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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.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 12:11, 16 March 2022 | | 480 × 480 (2.03 MB) | Berto (talk | contribs) | Uploaded own work with UploadWizard |
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