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Morphocular | Can you change a sum by rearranging its numbers? --- The Riemann Series Theorem @morphocular | Uploaded 3 years ago | Updated 3 hours ago
Normally when you add up numbers, the order you do so doesn't matter and you get the same sum regardless. And, of course, the same holds true even if you add up infinitely many numbers.....
Right?

=Chapters=
0:00 - Let's rearrange a sum!
1:48 - Investigation
6:32 - Riemann Series Theorem explained visually
13:58 - Resolving objections
18:52 - A step further and a challenge
20:07 - Significance of the Riemann Series Theorem
21:47 - Final thoughts

This video is a participant in the 3Blue1Brown First Summer of Math Exposition (SoME1). You can find out more about it here:
3blue1brown.com/blog/some1
#SoME1

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Want to support future videos? Become a patron at patreon.com/morphocular
Thank you for your support!

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The animations in this video were mostly made with a homemade Python library called "Morpho".
If you want to play with it, you can find it here:
github.com/morpho-matters/morpholib

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$How to Design a Wheel That Rolls Smoothly Around Any Given Shape Go to https://brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription. In previous videos, we looked at how to find the ideal road for any given wheel shape and vice-versa, but what about getting two wheels to roll smoothly around each other? Would two such wheels work as gears? Episode 1: https://youtu.be/xGxSTzaID3k Episode 2: https://youtu.be/Y0aOxj5lrKY =Chapters= 0:00 - Intro 1:23 - Defining smooth rolling 2:30 - Sidenote about gears 3:16 - The Wheel-Coupling Equations 7:34 - Sanity check 9:24 - The partner for an ellipse 12:24 - The connection between ellipses and parabolas 13:23 - Finding self-coupling wheels 16:35 - The partner for a square 19:21 - A look back 20:20 - A fractal wheel?? 20:47 - Brilliant ad I would also like to thank the user @BeekersSqueakers whose comment I think it was that taught me that a partner wheel can be generated by first generating a road and then generating a wheel on its underside. This comment was directly responsible for inspiring the technique shown in this video to easily generate self-coupling wheels, and dramatically simplified the second half of this video! So a seriously genuine thank you to @BeekersSqueakers and to all those who actually took up the call to answer my challenge problems in a comment! They can have a surprisingly big impact sometimes! For a deeper dive into the concepts explored in these videos, take a look at the paper Roads and Wheels, an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). You can find it here: https://web.mst.edu/~lmhall/Personal/RoadsWheels/RoadsWheels.pdf CREDITS The music tracks used in this video are (in order of first appearance): Rubix Cube, Checkmate, Ascending, Orient, Falling Snow The track Rubix Cube comes courtesy of Audionautix.com Want to support future videos? Become a patron at https://www.patreon.com/morphocular Thank you for your support! The animations in this video were mostly made with a homemade Python library called Morpho. If you want to play with it, you can find it here: https://github.com/morpho-matters/morpholib$

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Can you change a sum by rearranging its numbers? --- The Riemann Series Theorem @morphocular